I stole this idea from Andy, but I've seen quite a few less than him of these award winning movies. The ones in bold I've seen beginning to end, whereas the ones with an asterisk* I've only seen portions of. I've read the book versions of the ones in italics .
2006 The Departed
2005 Crash
2004 Million Dollar Baby
2003 The Lord of the Rings: The Return of the King
2002 Chicago
2001 A Beautiful Mind
2000 Gladiator
1999 American Beauty
1998 Shakespeare in Love
1997 Titanic
1996 The English Patient
1995 Braveheart
1994 Forrest Gump
1993 Schindler's List
1992 Unforgiven
1991 Silence of the Lambs
1990 Dances with Wolves
1989 Driving Miss Daisy*
1988 Rain Man
1987 The Last Emperor
1986 Platoon
1985 Out of Africa
1984 Amadeus*
1983 Terms of Endearment
1982 Gandhi
1981 Chariots of Fire
1980 Ordinary People
1979 Kramer Vs. Kramer
1978 The Deer Hunter
1977 Annie Hall
1976 Rocky*
1975 One Flew Over the Cuckoo's Nest
1974 The Godfather Part II
1973 The Sting
1972 The Godfather
1971 The French Connection
1970 Patton
1969 Midnight Cowboy
1968 Oliver!
1967 In the Heat of the Night
1966 A Man for All Seasons
1965 The Sound of Music
1964 My Fair Lady
1963 Tom Jones
1962 Lawrence of Arabia
1961 West Side Story
1960 The Apartment
1959 Ben-Hur
1958 Gigi
1957 The Bridge on the River Kwai
1956 Around the World in 80 Days
1955 Marty
1954 On the Waterfront
1953 From Here to Eternity
1952 The Greatest Show on Earth
1951 An American in Paris
1950 All About Eve
1949 All the King's Men
1948 Hamlet
1947 Gentleman's Agreement
1946 The Best Years of Our Lives
1945 The Lost Weekend
1944 Going My Way
1943 Casablanca
1942 Mrs. Miniver
1941 How Green Was My Valley
1940 Rebecca
1939 Gone With the Wind
1938 You Can't Take It With You
1937 The Life of Emile Zola
1936 The Great Ziegfeld
1935 Mutiny on the Bounty
1934 It Happened One Night
1933 Cavalcade
1932 Grand Hotel
1931 Cimarron
1930 All Quiet on the Western Front
1929 The Broadway Melody
1928 Sunrise
1927 Wings