When I get the chance I will update the build to fit the new AP item changes along with Runeglaive. I feel that the new Archangel Staff and Runeglaive will give this build more damage. Please be patient for future updates to come. I am waiting until the AP items are fixed/changed.

TYVM, GL&HF

Please note I am not calling Ryze jungle the new meta I'm saying that it is completely viable and fun if you know what you are doing.

PS: This is my first guide and I worked on it for a long time please go easy on me and give your opinion and recommendations in the comments. (If there are any spelling errors or changes let me know) GLHF :D

Ryze was one of the first champions I learned to play he was very fun and you could throw out a lot of abilities very quickly. At level 6 Ryze has amazing wave clear and can take out waves of minions very quickly, jungling before level 6 is hard to do although it is possible. The reason why I think Ryze would make a really great jungler is because of his ganking capabilites. Taking Ghost instead of flash and allows for a level 4 gank using his W (Rune Prison) which you really can't miss with. Ryze has high ganking capabilities and is somewhat tanky after level 6.

Why do i recommend these items?
[1] Stalker's Blade - Runeglaive | It gives you half the CDR you need and some ability power, it also makes ganking easier.
[2] Boots of Swiftness - Alacrity | It makes ganking easier and allows for better escapes although you can swap it out with Ninja Tabi or Mercury Treads.
[3] Seraph's Embrace/ Tear of the Goddess | Taking TOTG early gives you more mana and mana regen (as Ryze mana=damage).
[4] Rod of Ages | Taking Rod of Ages should be a no brainer, immediately after you buy TOTG you get ROA so you get more mana and health each minute.
[5] Frozen Heart | Taking FH is a good idea because it gives more mana (mana=damage) armor and the last bit of cooldown reduction you need.
[6] Zhonya's Hourglass/ Rabadon's Deathcap/ Abyssal Mask | The last item you take is really up to you. I recommend Zhonya's, Deathcap, or Abyssal Scepter. Zhonya's for the armor, ability power, and the 5 second invincibility. Abyssal Scepter if you need the extra magic resist instead of armor. I would advise Deathcap for the extra damage if you don't need the extra defense, deathcap is the better choice if your team is pretty tanky and you need to deal more damage.

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Arcane Mastery

His passive is whenever he casts 5 abilities he gains a mana sheild based on his level and maximum mana. Ryze's new passive allows him to be tankier and to take less damgage in the jungle or in fights.

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Overload

His Q is Overload a skillshot, all of Ryze's damaging abilities scale with AP and his maximum mana. Another thing to note is whenever he casts an ability the cooldowns on his other abilities are reduced by 1 second. Max this spell first.

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Rune Prison

Ryze's W Rune Prison is usefull for ganks, it holds the targeted enemy in place and deals some damage. Max this ability last, but take a point for early ganks.

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Spell Flux

Ryze's W is Spell Flux you throw a ball of magic that bounces back and forth in a circle area. Max this ability second so you can clear jungle or minion waves faster.

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Desperate Power

Ryze's ultimate ability Desperate power gives him alot of CDR and spellvamp, it also makes all your other abilities into a damaging AOE with 40% CDR your ult will have a 24 second cooldown and because you are ryze using abilities refreshes it faster, this will cause your ULT to be up very frequently. Don't be afraid to use it to get out of a sticky situation or chase because it will be back up very quickly.

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PS: If I do have time I will try to get my own screenshots of actual gameplay.

After you have 40% CDR you can jungle and wave clear really easily. When you use an ability as Ryze the cooldowns of his other abilites are lowered by one second, since riot changed his Q (Overload) to a skillshot it allows you to use it anywhere. So when your R (Desperate Power) isn't up you can throw out random Q's (Overload) to lower the cooldown of your R (Desperate Power). This is just an example if any of your skills are down and you want them back up you can now spam Q (Overload).

[*] Pros

+ Great ganking capability
+ Great farm past level 6
+ Great damgage
+ Great teamfight damage
+ Somewhat tanky
+ Somewhat fast
+ AP jungle
[*] Cons

- Slow early game
- Hard to jungle before level 6
- Hard to fight invades
- Not great 1 v 1

Your Jungling Route Should be as follows.
[1] Gromp | Smite Gromp immediately for the smite buff and finish it as fast as you can. Hopefully you have enough health and mana to fight blue buff, if your top lane or your support wouldn't mind taking a few hits it makes it much easier.

[2] Blue Buff | Use a health and mana potion as soon as Gromp is dead so you can regen some. Killing blue buff grants you CDR and mana regen it is a must.

[3] Scuttle Crab | Your smite shouldn't be up when you engage the Crab wait until you kill the Crab and have your smite cast.

[4] Murkwolves | Smite the wolves immediately and finish them as fast as you can, by now you should be level 3 Recall if you have enough gold to buy the jungle item do so if not buy 2 mana and 2 health pots.

[5] Red Buff | Smite red buff when you have low health.

[6] Scuttle Crab | If your smite is down go kill the Crab If not proceed to 7.

[7] Krugs | Smite as soon as you can and finish off quickly.

[8] Raptors | If you think you can kill the raptors go ahead although if any lanes have over extended you may want to gank.
VERY IMPORTANT: There are points in the jungle where a camp will stop chasing you it is very usefull to learn to manipulate this allowing you to take less damage in the jungle, however ranged jungle camps can still hit you and wont move they are the most dangerous.

Ryze has amazing potential for ganking with his W (Rune Prison) taking Alacrity, Ghost, and using R (Desperate Power) will allow you to dish out lots of damage and keep your target in place, using all these abilities will also help you chase.

Ryze Jungle is a lot of fun to play with your friends in casual but my final recommendation is down below. I don't recommend ranked play unless you are experienced enough as a jungle Ryze and have a team who knows how to play with you.