Showtime’s “Homeland,” which many critics have praised as the most riveting new drama of the television season, will reach its finale on Sunday night in an episode that the president for entertainment for the network, David Nevins, said “will have the fans of the show howling.”

He meant that in both a good and a bad way. “I imagine it’s going to be controversial,” Mr. Nevins said in a telephone interview. “It’s a good place to leave a series, with fans thinking, ‘Oh, my God, what are they possibly going to do now?’ The season definitely ends with a ‘what are they going to do now.’ ”

Already the most successful first-year series in Showtime’s history, with more than four million viewers watching each episode on cable and on demand, “Homeland” is a psychological thriller about an unstable C.I.A. officer, played by Claire Danes, and her obsession with a Marine P.O.W. (Damian Lewis), who she believes may have returned from Iraq as a Qaeda sleeper terrorist.

Sunday, after 11 episodes that have placed the characters on what could be a deadly collision course, the series faces a challenge that many recent gripping serial dramas have confronted: how to bring the season to a climactic end without leaving its most loyal fans “howling” in disappointment. “Lost” ended its six-season run on ABC with a finale that left many of its fans cold. The NBC series “Heroes” burned out quickly, adding confusing complications and not many resolutions each season.