Below is a list of significant 20th and 21st century composers. Each piece links directly to either a Naxos or YouTube recording of the work. While this list is by no means exhaustive, it does provide a starting point.
John Adams: (b. 1947)
John Luther Adams (b. 1953)
Milton Babbit: (1916-2011)
Grazyna Bacewicz: (1909-1969)
Samuel Barber: (1910-1981)
Bela Bartok: (1881-1945)
Alban Berg: (1885-1935)
Leonard Bernstein: (1918-1990)
William Bolcom: (b. 1938)
Pierre Boulez: (b. 1925)
Benjamin Britten: (1913-1976)
John Cage: (1912-1992)
Elliott Carter: (1908-2012)
Aaron Copland: (1900-1999)
John Corigliano: (b. 1938)
Henry Cowell: (1897-1965)
George Crumb: (b. 1929)
Claude Debussy: (1862-1918)
Morton Feldman: (1926-1987)
Other prominent composers of the 20th and 21st century:
George Gershwin: (1898-1937)
Philip Glass: (b. 1937)
Sofia Gubaidulina: (b. 1931)
Jennifer Higdon: (b. 1962)
Paul Hindemith: (1895-1963)
Lee Hyla: (1952-2014)
Charles Ives: (1874-1954)
Aaron Jay Kernis (b. 1960)
Erich Wolfgang Korngold: (1897-1957)
Ladislav Kubik: (b. 1946)
Gyorgy Ligeti: (1923-2006)
Witold Lutoslawski: (1913-1994)
Olivier Messiaen: (1908-1992)
Darius Milhaud: (1892-1974)
Conlon Nancarrow: (1912-1997)
Arvo Part: (b. 1935)
Kryzstof Penderecki: (b. 1933)
Francis Poulenc: (1899-1963)
Sergei Prokofiev: (1891-1953)
Maurice Ravel: (1875-1937)
Steve Reich: (b. 1936)
Terry Riley: (b. 1935)
Erik Satie: (1866-1925)
Alfred Schnittke: (1934-1998)
Arnold Schoenberg: (1874-1951)
Gunther Schuller: (b. 1925)
Ruth Crawford Seeger: (1901-1953)
Roger Sessions: (1896-1985)
Dmitri Shostakovich: (1906-1975)
William Grant Still: (1895-1978)
Igor Stravinsky: (1882-1971)
Karlheinz Stockhausen: (1928-2007)
Virgil Thomson: (1896-1989)
Joan Tower: (b. 1938)
Edgard Varese: (1883-1965)
Anton Webern: (1883-1945)
Ralph Vaughan Williams: (1872-1958)
La Monte Young: (b.1935)
Ellen Taaffe Zwilich: (b. 1939)