Computer accompaniment began in the eighties as a technology to synchronize computers to live musicians by sensing, following, and adapting to expressive musical performances. The technology has progressed from systems where performances were modeled as sequences of discrete symbols, i.e., pitches, to modern systems that use continuous probabilistic models. Although score following techniques have been a common focus, computer accompaniment research has addressed many other interesting topics, including the musical adjustment of tempo, the problem of following an ensemble of musicians, and making systems more robust to unexpected mistakes by performers. Looking toward the future, we find that score following is only one of many ways musicians use to synchronize. Score following is appropriate when scores exist and describe the performance accurately, and where timing deviations are to be followed rather than ignored. In many cases, however, especially in popular music forms, tempo is rather steady, and performers improvise many of their parts. Traditional computer accompaniment techniques do not solve these important music performance scenarios. The term Human-Computer Music Performance (HCMP) has been introduced to cover a broader spectrum of problems and technologies where humans and computers perform music together, adding interesting new problems and directions for future research.