I've decided to main Veigar(Vivi) since im a mega fan of Nuklearpower's Black Mage and am ecstatic to have his darkness at my fingertips. I used the layout of Kaneh's Tristana Guide. 

Veigar! Tiny Master of Evil!!!

HERO ANALYSIS

Veigar is a damage caster with very good AP scale to keep him useful throughout. He can quickly burst down 1 to 2 champions in teamfights with proper aiming. With his aoe stun and long range aoe hes not quite as fragile as most casters. Alot of fun to play since you are always seeing AP gain. 

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ABILITIES

Entropy (Passive): Veigar steals 20% of each nearby enemy champion’s Ability Power. The amount stolen per target cannot exceed five times Veigar’s level.

Take a few early levels of Baleful Strike to last hit creeps with so you can stock up on AP. A level of Event Horizon in order to do set ups and escapes. Then max out Dark Matter asap for farming. Primordial Burst does negligible damage early on so I take it when I'm 8 or 10 (sometimes ill level mid battle and take it earlier when i need a finisher.)

Summoner Skills:

Flash and Teleport: Flash can be used to chase or escape and everyone by now knows its pretty effective. Teleport happens to be my all time favorite cuz you can Blue Pill to base, buy then be right back in your lane fully refreshed in less than 20 seconds.


RUNES AND MASTERY TREE:

Runes: 
I suggest starting with at least 7-9 AP at the start. This will make DM hit high enuf to finish ranged minions at an earlier level. Use 10 spell pen and the rest in AP/18. 

You wont need the cd reduction if you have the 9% from masteries, 15% from shroud and 10% from brillianc pots. 

Ive had success maining the defensive tree as well but i had a much harder time keeping my mana up to continually cast baleful, so its up to you.


ITEMS:

Start with Meki Pendant and 2 Health Potions. 

On your first trip back get a Chalice of Harmony and if you can afford it a Ruby Crystal. 
Both of these are important for getting the most out of your early game. Ruby will save you countless times. When building Zhonya's i opt to sell it for the second Blasting Wand(40AP).



If the game is still going at this point... i usually sell my Chalice then Soul Shroud and get a Frozen Heartor Rylai's Scepter.

Note: once you have your Zhonya's feel free to get Brilliance Potions, they are a cheap way of increasing the +25% ap (65+16=81ap) and your cds further. I feel that getting them before this will set back your lategame since it'll take you about 3-4 waves to farm 1. Thx to Roku for bringing this to my attention.

Early game:

Play defensively! Do not engage enemy champions unless they come to you. Focus on Baleful Strike last hitting minions since you gain 1 AP for every kill. After you've used up both your heal pots or reach 500g and are low on mana Blue Pill back to base and buy your Chalice of Harmony. You may want another Heal pot or two. Once you get lvl 3 Dark Matter begin spamming it on the ranged creeps in every creep wave anywhere you go. Depending upon how much AP you acquired u may have to last hit each one. You'll make grotesque amounts of gold doing this and push enemy champions back.

Mid game:

Around this time is where you shine. You now have your Mejai's Soulstealer and its time to take some souls. Use your best judgment to place Dark Matters where enemy champions are then quick cast Event Horizon closer to you so that the outer ring stuns them. 

This may require some practice for precision stuns but is your bread and butter combo. If someone is chasing you Dark Matter so that if they keep coming they will have to take it to the face. Sometimes you can move forward with and ally then cast Event Horizon around them then cast Dark Matter in the direction they will be forced to retreat.

After that spamming Baleful Strike and Primordial Burst when necessary. [Remember] Finishing a Champion with Baleful Strike yields 5 AP so don't let opportunities pass up. 

Once you finish Soul Shroud you may want to buy a Brilliance Potion to pump up for some kills. You should get one after Shroud and after Zhonya's.

Late game:

Same as mid except team pushes should be happening more frequently. Event then Dark Matter. Primoridal Burst should be Hitting for insanely high amounts now even able to 1 shot certain Champions. Also once you acquire Zhonya's you should buy 300g Brilliance pots for an easy 81 AP and remember to kill Golems for the cd buff.


LAST THOUGHTS

This build has gotten me beyond Godlike in nearly every game ive used it and i hope it works for you on your way to learning the ends and outs of using Veigar. Hes alot of fun especially later when ive gotten his Dark Matter to hit for 960 and Primordial Burst for 400+760+25% of enemies mana. 

[Edit] Now that DM only takes 1.6 secs to land it doesnt have to be cast before EH. The other way is safer and about as effective.