This guide will show how Ezreal can be played, 3 different ways.
AD Ezreal(bot with support)
AD Ezreal(Mid)
AP Ezreal(Mid)
Ezreal is a strong hero with good escape abilites but no extra dmg on autoattack who makes his autoattack weaker than Cait or MF but he got skills that make up for that.

AD Ezreal
Pros - Best escape abilites of the range carries.
Good long range dmg.
Great ulti to finish them of.
Mystic Shot works with Trinity Force for some badass dmg.
Cons - No autoattack buff such as MF and Cait got.
Harder to play than the rest of the range carries(except Corki)

AP Ezreal
Pros - Good escape abilites.
High Burst.
Great finishing abilities(ulti).
Mystic Shot works with Lichbane for some badass dmg.
High aoe dmg.
Cons - No stun or slow(if no rylai).
Squishy.
Long cd on spells.

AD Ezreal, pick those runes because AD>Armor Pen imo because AD stack with Ulti and Armor Pen doesn't.

AP Ezreal, pick those runes because pure dmg and some armor for surv.

AD Ezreal
15/0/15 Because it give the extra movementspeed from utility but still extra dmg from offense.
21/x/x Works to but I really like the movementspeed.

AP Ezreal
9/0/21 so you get the regen and movementspeed from the utility and the magic pen from offense.
9/21/0 works for pure AP.
21/x/x works to but no movementspeed then.

AD Ezreal
Starting with Doran's blade if you lane with support and just wanna farm, if not you start on the Manamune and 2 hp pots so you can harass heroes and gank early.
Then Finish the boots or manamune depending on your farm, some go with the Ionan boots but I like the atk speed from Berserker and endgame you wont spell that much.
Get Sheen and upgrade to Trinity Force to have high harass with Mystic Shot.
I then get Last Whisper or Banshee's Veil depending on their team. Many casters Banshee, many tank Last Whisper, then I Black cleaver to get loads of dmg.
I finish my build with Bloodthirster to get high AD but some lifesteal to survive the 1v1 fights.

AP Ezreal
Doran's ring - Great start item
Sheen - To inrease Mystic Shot harass
Lichbane - To get the movement speed and higher burst
Rabadons - To get fully dmg fast
Void staff or Rylais - Depending on opposite team
Finish with Zhonyas to survive teamfights and then blink away

Another usefull build
Sapphire Crystal - To get Rod of Ages fast
Crystallize Protector - To be able to stay on lane and farm
Sorc boots - To gain that extra Magic Pen when you don't play Void Staff
Rod of Ages - Very good item that gives you hp/mana/ap
Archangel Staff - Increas your mana pool so you wont get oom and increas ap alot(very good with Rod of Ages)
Lichbane - You must have this item on AP Ezreal else you do to low dmg
Rabadons Deathcap - A must for all ap champs
Zhonyas Hourglass - To increas 100 ap and a surv item

I prefer the first build because it gives more dmg, the second build makes you surv better but you wont have enough magic pen to kill their tankish champions, you wont have Rylais neither and that makes you not be able to slow your opponoents.

AD Ezreal
Obvious because Mystic Shot is your dmg ability

AP Ezreal
I spec most in Essence Flux because it's highest dmg and aoe, and easiest to hit with.
I harass their mid with the flux when it's of cooldown getting him low and then blink - Combo - Ignite - Finish with ulti

Flash - Your blink spell got high cooldown so Flash will be needed

Ignite - To get that extra burst and to be able to 1v1 Mundo or WW

Other usefull spells :

Exhaust - Instead of Flash you can slow them and just autoattack, prefer on AD not AP

Clairyvoyance - Great with AP Ezreal to finish them with ulti

AD Ezreal
If you go mid with him you got 2 different ways to play, depending on who you meet, for example if you meet LeBlanc or Annie don't try to get FB, just farm because you will own mid game with a decent farm, but if you meet champions like Kassadin you should harass him with mystic shot and when he get 50% you do your combo "E"-"Q"-"Ignite"(If you aren't lvl 6).
If you go bot, you can play at 2 different ways, if you go with support you should just farm but if you go with a gankish teammate like Alistar you should do like you did mid against kassadin, harass and kill.

AP Ezreal
I start skilling "W" because it's a easy harass spell with high dmg, same here, don't try gank LeBlanc or Annie(If they got Ignite), Just farm.
If you meet Kassadin or a nuker without Ignite you should harass him with "W" all the time and when he gets to 50% you go "E"-"W"-"Q"-"Ignite" and you should have a kill.
You should not try to outfarm your opponent because AP Ezreal isn't very strong lategame, so go gank lanes very often because you can do that very easy because of your "E" spell.

AD Ezreal
If you have been mid you should run around gank all the time, push towers and remember to keep your passive stack up while you atk tower "W" at your teammates to do that and "Q" the enemies.
In a teamfight you should not "E" into it if you aren't outnumbering them, save it to the escaping or to finish someone of.
If you were on lane you should farm and farm, let your lanefriend run around and gank/help you can push your lane while your opponents are running around because with wards and your "E" spell you can escape rather easy.

AP Ezreal
No idea to farm now, lets push tower or gank.
In a teamfight you should stay behind and w8 for them to stay in a grp and get stunned, then "E"-"W"-"R" and you have done 1500 dmg to them then just mystic shot to finish them but don't "E" again if you aren't outnumbering them because you need it to escape.

Ezreal is a great champion who would be picked more often, all carries plays MF/Ashe/Cait(at my elo) but they forget Ezreal, Corki also, they both got spells and not just autoattack, champs like Cait just loose dps when spelling and Ezreal got better escape abilities.
Ezreal works as AP to but not as affective lategame but if you play with Amumu, Malp or any other aoe stunner you can do loads of dmg.

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