The capacity of the underwater acoustic communication (UAC) channel is addressed under a comprehensive set of assumptions: time-varying multi-paths channel, modeled as a doubly-spread frequency-selective Ricean channel with known statistical properties, actual realization of the channel unkown to the transmitter and receiver, constraints on transmit power(averaged and/or peak) and available frequency bandwidth. The exact channel capacity under such general assumptions is still unknown. Therefore, upper and lower bounds of this capacity are given, and then numerically assessed and discussed for a few typical shallow water UAC channels. It is shown that, as long as the theoretical channel capacity is considered, transmission with spectral efficiency higher than often now (e.g. 2 to 3 bits/sec/Hz) appears as a reasonable objective in typical UAC channels, providing SNR about 15 to 20 dB. In other respects, for a given available averaged transmit power, large peak-to-averaged-power-ratio (PAPR) is also shown to be highly desirable, since it significantly reduces the capacity loss due to the channel uncertainty.