Riven
Violence to End Violence
Lane harassment and consistent damage output.
Precision
Improved attacks and sustained damage
ConquerorGain stacks of adaptive force when attacking enemy champions. After reaching 12 stacks, heal for a portion of damage you deal to champions.
TriumphTakedowns restore 5% of your missing health and grant an additional 20 gold.
Legend: AlacrityTakedowns on enemies grant permanent Attack Speed.
Coup de GraceDeal more damage to low health enemy champions.
Sorcery
Empowered abilities and resource manipulation
TranscendenceGain bonuses upon reaching the following levels:
Level 5: +5Ability Haste
Level 8: +5Ability Haste
Level 11: On Champion takedown, reduce the remaining cooldown of basic abilities by 20%.
Gathering StormGain increasing amounts of AD or AP,
adaptive over the course of the game.
Adaptive Force
+9
Adaptive Force Adaptive Force
+9
Adaptive Force 
Health Scaling
+10-180 Health (based on level)
Why Conqueror
Conqueror's AD steroid is fantastic for Riven as she has generous ratios across all of her abilities and the extra True Damage makes you incredibly difficult to itemize against.
Why Triumph
Triumph's extra burst healing is fantastic for teamfights where you're able to burst the backline. The health with allow you to reset and continue dealing damage.
Why Legend: Alacrity
The extra bit of Attack Speed from Alacrity for perfect for assisting your ability to weave in auto attacks quickly between [Q] casts for a fast combo.
Why Coup de Grace
Coup De Grace's execution pattern of bonus damage synergizes incredibly well with Riven [R] and can help you snowball and 1-shot backline champions.
WhyTranscendence
As a mana-less champion, Riven is highly reliant on as much CDR as she can get her hands on. The Extra CDR in the mid-game assists you in getting a quick 40% while turning into additional AD for kill pressure in late game full-build situations.
WhyGathering Storm
Gathering Storm on Riven makes you an absolute late game monster, able to one-shot any backline champion with the right opening.
Sacrifices Must be Made
Lane harassment and consistent damage output.
Inspiration
Creative tools and rule bending
First StrikeWhen you initiate champion combat, deal 7% extra damage for 3 seconds and gain gold based on damage dealt.
Triple TonicUpon reaching level 3, gain an Elixir of Avarice.
Upon reaching level 6, gain an Elixir of Force.
Upon reaching level 9, gain an Elixir of Skill.
Cash BackGet some gold back when you purchase Legendary Items.
Cosmic Insight+
18 Summoner Spell Haste
+10 Item Haste
Sorcery
Empowered abilities and resource manipulation
- Transcendence
Gain bonuses upon reaching the following levels:
Level 5: +5Ability Haste
Level 8: +5Ability Haste
Level 11: On Champion takedown, reduce the remaining cooldown of basic abilities by 20%. - Gathering Storm
Gain increasing amounts of AD or AP,
adaptive over the course of the game.
Adaptive Force
+9
Adaptive Force Adaptive Force
+9
Adaptive Force Health Scaling
+10-180 Health (based on level)
Why First Strike
Klepto on Riven synergizes well with your ability spam combos, allowing you to generate a lot of extra gold during the early laning phase.
Why Triple Tonic
The stasis active allows you to make out-plays and survive a close tower dive to snowball your lead. It also grants you a cheaper Guardian Angel.
Why Cash Back
Future's market lets you hit you item power-spikes extra early. When combined with Lethality, your mid-game damage will be incredibly high.
Why Cosmic Insight
Riven's mana-less kit means that CDR is one of her absolute most important stats. The extra bit of Summoner CDR also means more Flash and Ignite pressure.
WhyTranscendence
As a mana-less champion, Riven heavily relies on ability spam to deal damage. More CDR means more casts and more opportunities to kill.
WhyGathering Storm
The bonus AD granted from Gathering Storm makes you a late-game monster, able to one-shot any backline champions with the right opening.