Laurence Rosenthal composed the Young Indiana Jones theme used in the television broadcast opening credits and featured in many episodes, as well as scoring and conducting the music for just under half the episodes. Rosenthal's scores tend towards the "mellow" end of the spectrum. Rosenthal's home page lists his contributions to Young Indiana Jones and the awards he won for the series, plus several sample tracks for listening.
Joel McNeely also scored and composed for just under half of the episodes. His contributions tend towards "action" themes. Young Indy Around the World (a "making of" documentary) contains a segment (3 minutes in) where Joel McNeely talks about the music. BSOSpirit has an extensive interview with McNeely which contains a wealth of detail about his work on Young Indiana Jones. Highlights include:
Frédéric Talgorn composed the score for the Somme 1916 and Germany 1916 episodes that were later combined into Trenches of Hell. He also composed an unused score for Paris 1908. See his biography for some quotes about the experience. Talgorn's official Web site includes extensive samples from his body of work, including 3 tracks from Trenches (7 minutes of music).
Curt Sobel composed the chilling score to Transylvania 1918. His IMDb page lists his many accomplishments, and he shares his thoughts on working in the film industry in an interview with The Editors Guild Magazine.
Steve Bramson composed the score for Treasure of the Peacock's Eye. An online interview with Bramson details his inspirations, including Williams and Rosenthal. He has some samples of his work - including Peacock's Eye - at his home page. An interview with UNH (his alma mater) provides details about his family and many accomplishments, including an Emmy for Steven Spielberg's Tiny Toon Adventures and multiple nominations for the TV series JAG.
Episode 235 of The IndyCast features an interview with Steve Bramson by Mike Jozic where they discuss scoring the episode in detail.
John Williams' "Desert Chase" theme from "Raiders" (with the familiar cues) was used in the Mystery of the Blues bookends featuring Harrison Ford. The music was arranged by Joel McNeely with the rest of his score. It was all newly recorded for the episode.
Other composers' work is incorporated into the episode scores:
In addition, several episodes contain performances of music. The aviators sing "Garryowen" in Attack of the Hawkmen, both The Scandal of 1920 and The Mystery of the Blues feature performances by real and fictional characters, and even "China Dreams" in Treasure of the Peacock's Eye is a period song.