The traveling-wave activity that Georg von Bekesy studied in the cochlea early in this century has been of intense interest to physiologists and mathematical modelers over the last 30 years. This interest was nurtured by the discovery that the mechanical responses of the basilar membrane are nonlinear at relatively low sound levels and that the nonlinearity is dependent on biological processes [W. S. Rhode, J. Acoust. Soc. Am. Suppl. 1 49, S1218 (1971)]. This review will briefly summarize the major findings of the early years as background to a presentation of several issues that are the focus of contemporary work. The steady-state and transient velocity responses of the basilar membrane characterize the capacity of the biological amplification mechanism in the organ of Corti and indicate that a ``gain'' of 40--60 dB is provided by the outer hair cells. Considerable response distortion accompanies this gain, and wave propagation of certain distortion products occurs. Electrical stimulation of the outer hair cells provides evidence of their mechanical role in the amplification mechanism by demonstration of the frequency range and displacement capacity of the cells. Efferent innervation of the organ of Corti is found to modulate the mechanical activity of the system.