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Fuqua attained his Actor's Equity card after a two year internship with The Cincinnati Playhouse in the park (this after three years at James Madison University). Fuqua received his graduate level training at Yale Drama School and moved back to New York City upon graduation. He joined The Cuccaracha Theater of Soho and spent six years as a company member.
He is perhaps best known for his role as J.E.B. Stuart in the films Gettysburg and Gods and Generals. Fuqua also appears in the films David Searching opposite Camryn Manheim, Heyday, Something Else, and has an uncredited appearance as Bond Trader Boyce, a stockbroker, in Ed's Next Move.
He has guest starred in such popular television series as The X-Files (in the episode "Synchrony"), Star Trek: Deep Space Nine (in the episode "Rocks and Shoals"), Chicago Hope (in the episode "Gun with the Wind"), Becker, NYPD Blue, Beyond Belief: Fact or Fiction, The Profiler, G vs E, Brooklyn South and The Wright Verdicts. Joseph also appeared in the pilots for Second Nature, Born and Bred,Elementary and Numb3rs.
Fuqua has acted in over 100 plays over the past twenty-nine years, and has the honor of being the first permanent company member at the prestigious Rubicon Theatre Company in Ventura, California. He has acted in twenty-nine plays at Rubicon over the course of fifteen seasons (with his 30th appearance slated for Fall, 2014, in their 2013-2014 season). Fuqua received an L.A. Ovation Award (Rubicon's first-ever Ovation Award) for his performance as George in Arthur Miller's All My Sons, directed by Rubicon Artistic Director James O'Neill.
Throughout his tenure at Rubicon, Fuqua has played a variety of characters of different ages, types, dialects, and backgrounds. For Rubicon's 2006–2007 Balancing Acts season, Joseph played Hamlet, under the direction of Rubicon Artistic Associate Jenny Sullivan, for which he received the 2007 Indy Award for Best Actor and the 2007 Santa Barbara Independent Award Leading Actor Award.
Along with acting at Rubicon, Fuqua also is an instructor for their Adult Acting Classes, Fearless Shakespeare Youth Summer Program, does private coaching, served as Associate Director for their production of Jenny Sullivan's original play, J for J, and served as the play's director for the Los Angeles run starring John Ritter. He has directed the Youth Intern production of: This is Our Youth, and for Rubicon's Fearless Shakespeare Youth Summer Program, the Summer Youth productions of: A Midsummer Night's Dream, Bard On A Wire, Macbeth, As You Like It The Life and Death of King John, and Romeo and Juliet. In June, 2014 Joseph will direct King Lear for that program.
Fuqua's stage appearances outside of Rubicon include: The Mystery of Irma Vep, Private Lives and Six Dance Lessons In Six Weeks (Ensemble Theatre, Santa Barbara, CA), and Fabuloso (Wellfleet Actors Harbor Theatre, Wellfleet, MA). Other regional theatre credits include The Arena Stage, Portland Stage, Dallas Theatre Center, Actor's Theatre of Louisville, South Coast Repertory, Santa Fe Stages, New Jersey Shakespeare Festival, The Dallas Shakespeare Festival, The Manitoba Theater Center, The Edinburgh Festival, Studio Arena Buffalo, The American Stage Festival, People's Light and Theater, Cincinnati Playhouse and The Yale Repertory Theater.