Runes

SORCERY

Arcane Comet
is one of the best runes to take on Lux.It hurls a comet towards the enemy when you hit them with an ability. since all of her basic abilities are skill shots this rune is realy good on her.

Manaflow Band
is a very handy rune for Lux. her abilities have high mana costs. taking Manaflow Band gives you 25 mana every time you hit an enemy champion with an ability. This rune is very good in your early game.

Transcendence
grants you 10% cooldown reduction at level 10, which means that you Will have a lower cooldown on your abilities this rune is useless in early game but helps very much in the mid game and late game because of team fights.

Gathering Storm
is also useless at the start of the game but pays off at the mid game and late game which gives you bonus Ap every 10 minutes.
Runes

PRECISION

Presence of mind
allows Lux to always have mana during team fights and also decreases the cooldown of her ult. It's also Ties up all of your rune build together.

Coup de Grace
Coup de Grave gives you free damage to finish low Hp enemies. and Because Lux most of the time with her ultimate this bonus damage is very useful in any part of the game.
Runes

INSPIRATION

Magical FootWear
gives you bonus movement speed and helps you catch up with enemies running away from you or the opposite however you wouldn't be able to buy boots until later in the game which isn't a problem because you build before buying boots so enough time will already pass to buy upgraded boots

Cosmic insight
gives you bonus CDR and personally i like to get it because I like to poke with Lux and if she has even less cooldowns you can poke a lot harder. It makes a big difference
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Core Items

Doran's Ring
gives you extra mana,HP and AP. What else did you ask for?

Luden's Echo
Very Good Core Item which gives you a lot of cooldown reduction,More AP and makes it a lot easier to CS.

Sorcerer's Shoes
Very good to deal extra damage to enemy champions due to the magic resistance reduction.

Rabadon's Deathcap
Gives you a massive increase in Ability power which increases every time you get more ability power. cost a lot of gold but surely worth it.

Zhonya's Hourglass
Zhonya's Hourglass is good for reducing damage taken and a life saving passive which can deny ultimates like or .

morellonomicon
gives you 300 bonus health but the 70 AP and the 15 Magic Penetration it gives you is the reason it's important to buy this Item

void staff
Gives you a huge amount of magic penetration and a good last Item to buy.
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Situational Items

Banshee's veil
build this against Heavy AP Champions like . and when you play against teams with a lot of ability power champions.

Mejai's soulstealer
good if you are far ahead but if you die a lot it's not worth it.

Liandry's torment
build this only when you are against a team with a lot of tanks and high health bars.

Twin Shadows
gives you a good amount of AP movement speed and cooldown reduction but buy this item only when you see that your team has bad vision in late game. and also to check if there are any enemies who are solo so your team can ambush them.

spellbinder
Gives you Huge amounts of AP.buy this item if you feel like you don't have enough damage because the enemies are to tanky or you just don't have much damage.

Rylai's crystal scepter
you don't usually see this is item on Lux because there are better similar options for her like but because of it having a lower gold cost it also can be good sometimes.

Lich bane
Gives a good damage bonus for your basic attacks and works well with your passive. buy this when your team has a lot of CC so it will be easier for you to hit basics.
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Illumination
Illumination is Lux's Passive ability which marks the enemy with light for 6 seconds. Your next basic attack deals extra magic damage every time you hit an enemy with an ability: Light Binding, Lucent Singularity or Final Spark

Light binding
Light Binding roots up to two targets at once for 2 seconds and deals a good amount of magic damage to enemies hit. It's Lux's main CC ability.

prismatic barrier
Prismatic Barrier is Lux's shield ability.It travels as a boomerang and when used Lux throws her wand and shields her self and any other ally it meets, after Lux's wand starts to return it shield double the amount to any ally it interactes with.

Lucent singularity
Lucent Singularity is Lux's main Poke ability. When used: it leaves a field which slows any enemy within the field. If cast again it detonates the field and deals magic damage to enemies.

Final Spark
Final Spark is Lux's Ultimate ability. Final Spark is a high range and a low cooldown ultimate which makes Lux to charge haft a second a massive beem of light and deal huge amounts of damage to any enemies hit.