Konrad "Conny" Bauer (born 4 July 1943) is a German free jazz trombonist. He is the brother of the trombonist Johannes Bauer.

As a student at senior high school in Sonneberg between 1957 and 1961, he was enthusiastic about modern music and dance genres such as swing, boogie-woogie, blues and rock 'n' roll, and taught himself to play guitar and piano. After leaving school with A-levels, he tried to play his music in several bands and was nicknamed "Conny" by his friends.

After recognizing that he did not know enough about music to become a professional musician, Bauer studied modern dance music from 1964 to 1968 at the Carl Maria von Weber-Music school Conservatory in Dresden. Because too many students wanted to study guitar, he entered the trombone class, having had some experience of playing the instrument. In 1968 he left the conservatory for Berlin to improve his skills with private lessons. From 1969 until 1971 he started his career as guitarist and singer in the band of Ernst-Ludwig Petrowsky. He also began a career as a trombone soloist in 1970.

During the second half of the 1970s Bauer became a prominent jazz player in European free jazz. He helped found numerous groups which influenced the development of jazz in East Germany: these included FEZ and its successive quartet and trio formations, the Doppelmoppel quartet, and Synopsis/Zentralquartett with Ernst-Ludwig Petrowsky, Ulrich Gumpert and Günter "Baby" Sommer. He was featured in a profile on composer Graham Collier in the 1985 Channel 4 documentary 'Hoarded Dreams'. In 1986 he toured Japan for several weeks, encountering numerous Japanese musicians. From 1988 to 1989 he directed the National Jazz Orchestra of the former East Germany. Since 1983 he has worked with artists such as Tadashi Endo, Sheryl Banks, Tony Oxley, Derek Bailey, Maggie Nicols, Theo Jörgensmann, Peter Brötzmann, Barre Phillips, Peter Kowald, Han Bennink, Barry Altschul, Jay Oliver, Louis Moholo, Gerry Hemingway, and George E. Lewis.

In 2004 Bauer was awarded the German SWR jazz prize especially for his solo recordings Hummelsummen. As unaccompanied soloist Bauer uses multiphonics, with matchless "circular breathing techniques" he conjures his own loops.