Avatar won 3 Oscars in the 82nd Academy Awards, out of 9 it was nominated for. The film won a total of 65 (out of 128 nominations).
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Academy Awards:
Best Art Direction
Best Cinematography
Best Visual Effects

Art Directors Guild Awards:
Excellence in Production Design - Fantasy Film
Austin Film Critics Association:
Top 10 Films

British Academy Film Awards:
Best Production Design
Best Special Visual Effects

Broadcast Film Critics:
Best Action Movie
Best Art Direction
Best Cinematography
Best Editing
Best Sound
Best Visual Effects

Dallas-Fort Worth Film Critics Association:
Top 10 Films

Empire Awards:
Best Film
Best Actress
Best Director
Florida Film Critics Circle:
Best Cinematography

Golden Globe Awards:
Best Director
Best Film - Drama

Golden Reel Awards:
Best Sound Editing - Best Sound Editing: Music in a Feature Film
Best Sound Editing - Best Sound Editing: Sound Effects and Foley in a Feature Film

Las Vegas Film Critics Society:
Sierra Award (Best Art Direction)
Lumière Awards:
Live Action 3D Feature
Best 3D Character
Best 3D Scene
Best 3D Stereography / Live Action
Outstanding Achievement in 3D Visual Effects
Outstanding Achievement for Marketing 3D Content / Live Action

New York Online Film Critics:
Best Film
Top 11 Films

Oklahoma Film Critics Circle:
Best Ten Films

People's Choice Awards:
Favorite 3D Live Action Movie
Favorite 3D Animated Movie

Phoenix Film Critics Society:
Best Cinematography
Best Film Editing
Best Production Design
Best Visual Effects

PETA's Proggy award:
Outstanding Feature Film

Santa Barbara International Film Festival:
Lucky Brand Modern Master Award

Saturn Awards:
Visionary Award for Avatar
Best Science Fiction Film
Best Actor
Best Actress
Best Supporting Actor
Best Supporting Actress
Best Director
Best Writing
Best Music
Best Production Design
Best Special Effects

St. Louis Gateway Film Critics Association Awards:
Best Visual/Special Effects
Most Original/Innovative or Creative Film

Teen Choice Awards:
Sci-Fi Movie
Actress Sci-Fi
Actor Sci-Fi

Visual Effects Society Awards:
Lifetime Achievement, for Making of
Outstanding Visual Effects in a Visual Effects Driven Feature Motion Picture
Best Single Visual Effect of the Year, for Neytiri Drinking
Outstanding Animated Character in a Live Action Feature Motion Picture, for Neytiri
Outstanding Matte Paintings in a Feature Motion Picture, for Pandora
Outstanding Models and Miniatures in a Feature Motion Picture, for Samson, Home Tree, Floating Mountains, & Ampsuit
Outstanding Created Environment in a Feature Motion Picture, for Jungle / Biolume

There's one pending Hugo award for “Best Dramatic Presentation, Long Form”