Support Morgana is a lot like Support Lux. If you're comfortable with one, you should be able to play the other relatively easily. Morgana appeals to the good-girl supports who wish they could be bad.

...that and gingerbread.
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In my opinion, support Morgana relies HEAVILY on team composition. She works well with heavy-AP teams or heavy-CC teams. She will not work well with an AD-based team. She counters most bottom lane supports with her Black Shield, making her a good choice AGAINST CC-based teams.
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Music to listen to? idk. Filler space.

PROS Cuz we all pro + Black Shield makes your carry un-CCable. + Great harass + Most people don't expect her as a support + Synergizes very well with AP teams + Great split-pusher

CONS You thought you could get away? - No heal - Black Shield does not block physical damage - Her passive spell vamp doesn't really suit a support - Squishy - Difficult to handle her damage output

Soul Siphon is Morgana's passive, one of the reasons why she is generally considered a mid-laner. It grants you 10-20% spell vamp based on level. For a support, you're not concentrated on doing damage, so it makes it difficult for you to make full use of this passive. The only time I would consider it would be if your carry is missing and you want to push the lane, you can use Tormented Shadow to hit the minions and drain them.

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Your snare. It's a skillshot, but unlike Lux's Light Binding, it only hits the first target. It starts off as a 2-second snare and goes up to 3 seconds as you upgrade it. Beware! Snare does not mean stun. I train this skill first to get the snare damage and duration up.

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A debuff. Tormented Shadow creates a small area (which you can select) where enemies take damage over time and lose magic resistance. It makes for great harass and you can use it to convince enemies to move in a particular direction, like a trap. I train this last because its better utilized as a warning against movement, and your carry does not utilize the MR-debuff effectively.

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This is what makes Morgana so annoying. Black Shield blocks a certain amount of magic damage (with 70% AP scaling...not like we're building AP) and prevents all CC on the shielded ally until the shield is broken or disappears after 5 seconds. I train this second to increase the power of the shield and decrease its cooldown, as enemies should be wandering by now.

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Soul Shackles is a 3-second slow-leash that targets the two closest enemy champions. If players stay within its range for the full 3-seconds, they are stunned for 1.5 seconds.

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Throw a Tormented Shadow behind them, then shoot a Dark Binding as they run away to lock them onto the affected area. The initial hits of Tormented Shadow will also lower their MR to increase the damage done by Dark Binding.

Gold per 10 quints for extra gold. Supports aren't supposed to rely on making money via kills, though you certainly get a lot from assists. Do not farm minions unless your carry tells you to or your carry is not present, in which case you aren't stealing their creep score.

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Flat armor yellows for additional early game sustain. You can also use scaling health or scaling MR, but most of the damage you're taking will come from the enemy carry. Kills early game determine late game, so we wish to discourage this as much as possible. You can also take Greater Seal of Scaling Health or Greater Seal of Scaling Magic Resist, but I think they are situationally more effective.

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Magic Pen reds to increase your harassment damage.

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Flat CDR blues to decrease the CDR on Black Shield. These runes will be most disputed - if you disagree, please please PLEASE tell me a better option in the Discussion section. I was considering Greater Glyph of Scaling Ability Power to increase the effectiveness of Black Shield and her harassment, but I'm afraid of doing too much damage.

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I prefer Ionian Boots of Lucidity for the cheap 15% CDR. However, Mercury's Treads for Tenacity (when you need to Black Shield an ally) and Boots of Swiftness for keeping enemies in range of Soul Shackles are also viable. Enchantments are situational, but Enchantment: Alacrity is always a safe choice.

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Ruby Sightstone = Wards. "Free money"...you're not spending on wards. There is something seriously wrong if you are not warding your lane. Also, gives HP. Not that bad.

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Runic Bulwark is THE support staple. It's considered one of the most cost-effective items in the game. Armor and MR for not just you, but your allies (plus you get the aura buff also).

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Shurelya's Battlesong is another support staple. 10% CDR and a 3s 40% movement speed buff, along with a few other things. Builds off of Philosopher's Stone, so it makes a good endgame item.

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Twin Shadows builds out of Kage's Lucky Pick, which is a good early game item for a support. Specifically, the active from Twin Shadows is beneficial, especially when you use it in conjunction with Soul Shackles.

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Zeke's Herald

+20 AD and +10% lifesteal to all nearby allies! Your carry will love you, and it slightly makes up for your lack of sustain. I only recommend this on a high AD team.

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20% CDR and 95 armor, which gives Morgana a bit of tankiness. It also assists in discouraging attack speed champions, such as the enemy carry.

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Abyssal Mask is a situational pick. If your team is heavy AP and no other team member takes it, this is a good choice. It works well with Morgana's Soul Shackles yet again, slowing the enemy and giving them an MR-debuff as your team drops nukes on them. Don't forget Tormented Shadow!

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Hey look! Morgana's Black Shield is down! CC the enemy carry! ... HOW THE HELL DID HE ESCAPE? :D

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This is not your typical support item; in fact, I think the only champion who goes out of his way to build it is Heimerdinger. However, this is a good choice if you want to split-push. Combined with Morgana's natural wave-clearing abilities, this item offers armor, ability power, and 10% CDR.

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Zhonya's Hourglass is a typical pick for mid- Morgana. It provides the early Seeker's Armguard against AD-mids (in your case, AD carries) and then builds into an item with very high AP. Additionally, it's a popular tactic to show up behind the enemies (via Flash), use Soul Shackles on a heavily CCed enemy team, and then use the Hourglass's active to become immune while the timer for Soul Shackles ticks.

Early game, you fulfill a typical support role. Ward your lane and protect your carry. Morgana is a great harasser at early levels, so don't be afraid to use Tormented Shadow alone to harass, especially if you go for Chalice of Harmony. Placing it in the enemy bushes is pretty effective for weeding out their support, especially since it won't damage minions too often.

These are good places to put wards to protect yourself from ganks early game. Shaco, Shyvana, Nocturne, and Rammus tend to gank early game.
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Harass opponents until your carry is ready to kill them. When your lane is ready to push, start using Tormented Shadow to kill the caster minions while your carry takes his or her pick from the melee minions.
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After the first turret is down, you have two options. Stay in lane with your carry and keep pushing up bottom lane or wander with your carry. I recommend stealing the enemy buff (blue for Morgana or red for your carry) or at least warding that bush so you can see the buff monster. Lay a ward near the river entrances again to protect yourself from incoming sandwiches.
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If you go for wandering (most likely to mid, unless you recall to go to top), here's a reference for shooting Dark Binding from behind walls.
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It's pretty far, right? Of course, one should never shoot a skill shot perpendicular to the target's path, as it has the least chance of landing. Missing means you give away your presence. Landing...means your carry can do what carries do, provided they are in a position to engage.
████████████████████████████████████████████████████████████████████ By now, there should be a lot of team fights going on. This is where you excel, given how far you are in your build and your team composition. First things first: you want to keep dropping wards, especially near the most popular team fight lane (this is normally mid, but it's not unusual for it to be bottom or top).

Regarding team fights, you need to learn how the enemy team works. There are two lines of thought in team fights - target the carry (AD or AP) or target the closest (tank). It's is absolutely imperative that you learn how the enemy team works during the first team fight so you know who needs Black Shield during the next team fights. This is also why it's useful to have Mikael's Blessing; you can Black Shield the carry and prepare Mikael's for your tank (since it has a heal attached).

Other than that, focus on dropping your Tormented Shadow in the middle of the enemy team and use Dark Binding to catch escapees. Only use Soul Shackles when you want to polish off what's left of the enemy team. It's NOT a useful initiation tool because Morgana is easily killed, and now your team no longer has your aura benefits.
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Okay, go on. Use Tormented Shadow to clear the minions around the nexus turrets so your super minions have full reign. Lock out any enemy champions who come out with a Dark Binding. Don't get too close to the enemy platform, as you are still a squishy witch.
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Um. Hmmm. I would say your team probably wants you defending your base from super minions. Try to Soul Shackles the enemy team in your nexus turrets if possible. If you get any relief from the siege, stock up on Elixir of Brilliance and make sure your tank/bruiser gets Sight Wards and goes with your carry to the enemy base to take out inhibitors while the rest of your team tries to push the lanes back out.

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Ashe improves crit chance while out of combat. Has a slow, AoE, and passive gold bonus. Sync: Low. She doesn't have a strong early game presence and your ultimates are very similar.

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Caitlyn does extra damage every 2-3 AAs. Has a snare, linear AoE, and positioning mechanic. Ulti is almost inescapable. Sync: Low. Caitlyn is an overall strong pick, but she does not have any particular synergy with Morgana.

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Corki does 10% AD in true damage on AAs, but low range. Skills are damage-based. Sync: Medium. Has decent burst and AoE damage. Has no synergy with Tormented Shadow, but appreciates Dark Binding and Black Shield.

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Draven gets bonus crit damage. Has a damage skill, a speedup, and a positioning skill. Overall, better AA damage than other champions. Sync: Medium. Your snares are useful for him to max out his DPS, and you cover his gigantic weakness to CC via Black Shield.

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Ezreal gains attack speed after using abilities. Lots of ranged skills and a flash in his skills. Sync: Medium. You have synergy with his hybrid skillset, and the snares are great for locking down opponents for easy skillshots on his behalf. Run this lane as a poke AND burst lane.

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Graves gains armor & MR in combat. High utility kit (AoE, AoE slow/vision reduc, dash). Sync: Low. He has nothing to gain from laning with Morgana.

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Kog'Maw turns into a walking bomb on death. Has superior range and a slow. Sync: High. One of the matches made in heaven for Morgana. You both weaken the enemy's magic resist and Kog'Maw does magic damage. You can cover his escapes with Black Shield and Dark Binding. I'd give this pair an 11/10.

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Miss Fortune has improved movement speed while out of combat. Works well against healing teams. Sync: Low. Does not have particular synergy. Probably work well for a push-comp or a team-fight comp, but nothing special.

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Sivir gains movement speed with each AA. Has decent AoE damage and great chasing capabilities. Sync: Low. She can chase, so she doesn't really need your CC; she'd rather have a damage buff, which you can't provide.

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Tristana gains AA range as she levels. Specializes in positioning and AoE damage. Sync: High. Tormented Shadow → Dark Binding → Rocket Jump → Buster Shot (towards Morgana) → Soul Shackles → Explosive Shot. GG.

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Twitch does bonus DoT true damage every AA. Has a burst that builds off his AAs. Sync: Low. Does not benefit from Morgana at all.

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Varus gains a temp AS buff with each kill/assist. Focus of his damage is Blight, which does damage based on % health. Sync: High. The other match made in heaven. Both deal magic damage, have strong magic-based AoEs, snares set up for easy kills, and if that fails, Piercing Arrow. Delicious.

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Vayne gains a movement speed bonus while chasing enemies. Has a gap closer and positioning skill. Sync: Low. No particular synergy.