Millimeter wave (mmWave) communication is a promising technology for 5G cellular systems. To compensate for the severe path loss in mmWave systems, large antenna arrays are generally used to achieve significant beamforming gains. However, due to the high hardware and power consumption cost associated with radio frequency (RF) chains, it is desirable to achieve the large-antenna gains, but with only limited number of RF chains for mmWave communications. To this end, we study in this paper a new lens antenna array enabled mmWave MIMO communication system. We first show that the array response of the proposed lens antenna array at the receiver/transmitter follows a "sinc" function, where the antenna with the peak response is determined by the angle of arrival (AoA)/departure (AoD) of the received/transmitted signal. By exploiting this unique property of lens antenna arrays along with the multi-path sparsity of mmWave channels, we propose a novel low-cost and capacity-achieving MIMO transmission scheme, termed \emph{orthogonal path division multiplexing (OPDM)}. For channels with insufficiently separated AoAs and/or AoDs, we also propose a simple \emph{path grouping} technique with group-based small-scale MIMO processing to mitigate the inter-path interference. Numerical results are provided to compare the performance of the proposed lens antenna arrays for mmWave MIMO system against that of conventional arrays, under different practical setups. It is shown that the proposed system achieves significant throughput gain as well as complexity and hardware cost reduction, both making it an appealing new paradigm for mmWave MIMO communications.