Underwater acoustic (UWA) channels are typically characterized by a multipath structure with large delay spread, where only a few propagation paths carry significant energy. Each path exhibits time variability, which, together with the transmitter and receiver motion, induces Doppler spreading and shifting of the signal. In this paper, we analyze the limits on the information rate achievable through multiple-input multiple-output (MIMO) communications over UWA channels. Assuming full channel state information (CSI) at the receiver, we evaluate the ergodic capacity in two scenarios: one with partial CSI at the transmitter, and another with no CSI. Also, we consider the constrained capacity for practical modulations, e.g., BPSK and QPSK, and, exploiting the sparseness of the multipath structure, we provide new lower bounds on the achievable information rate. Statistical characterization and numerical examples are given based on the data collected in a recent experiment, conducted off the coast of Kauai, Hawaii, in June 2008.