The evolution of a mathematical model designed to clarify the relationship between the structure and the function of the cochlea (inner ear) is traced. Starting from physical (rather than empirical) considerations, the mathematical description of each model is derived in terms of well-defined physical properties of the cochlea. Initially, the basilar membrane is modeled as a uniform simply supported beam vibrating in a viscous medium, and driven by a concentrated oscillating moment at the basal end. In such a system traveling and standing waves (transients) occur simultaneously. An analysis of the traveling waves reveals a place principle. The model has high and low frequency "thresholds" (i.e., at very high and very low frequencies, frequency discrimination diminishes). The model suggests that the fluid properties play a particularly important role at high frequencies, while the role of cochlear geometry becomes dominant at low frequencies. In the second stage, the beam is enclosed in a rectangular "cochlea" divided into two equal chambers and filled with viscous and incompressible fluid (the perilymph) communicating through a small opening at one end (the helicotrema). At the opposite end, the system is driven by a piston-like periodic forcing (i.e., the movement of the stapes in the middle ear). It is assumed that the oscillations of the basilar membrane have long wavelengthmcompared with the cochlear chamber heightmand that a thin unsteady boundary layer forms on all surfaces within the cochlea. In general, the results confirm and improve the results of the previous model. The place principle exhibited by this model corresponds well with experimental data at higher frequencies, but not at low frequencies. For good low frequency response, the model must be endowed with much of the cochlea's intricate geometry. This is done in the third model where the basilar membrane is represented by a wedge-shaped isotropic plate of constant thickness and enclosed by an arbitrary surface of revolution representing the cochlear shell. The plate is simply supported along its long edges while three different boundary conditions are considered for the support at the helicotrema. So as to emulate the presence of the cochlear duct (third cochlear chamber), a pressure difference in the fluid at the helicotrema is included. As in the second model, the cochlear fluids are taken to be viscous and incompressible, and the system is driven by periodic forcing at one end. The solution of the equations of motion as the input frequency tends to zero is obtained. The role of the basilar membrane taper, as well as cochlear cross-section's opposing taper, is delimited. The third chamber (cochlear duct) must be included in the model in order to utilize the full length of the basilar membrane for frequency discrimination. Consequently, the boundary conditions of the basilar membrane at the helicotrema can be analyzed, and it is shown that an elastic support (representing a "ligament") is required at the helicotrema.