Exploiting massive multiple-input-multiple-output (MIMO) gains come at the expense of obtaining accurate channel estimates at the base station. However, conventional channel estimation techniques do not scale well with increasing number of antennas and incur an unacceptably large training overhead in many applications. This calls for training designs and channel estimation techniques that efficiently exploit the physical properties of the massive MIMO channel as captured by sophisticated system/channel models. In this paper, we present designs that exploit the sparsity of the angle and delay domain representation of the massive MIMO channel as well as the low-rank property of the channel covariance, while also providing the connection between the sparse angle-delay representation and low-rank covariance property. Numerous multiuser scenarios are investigated including uplink, downlink, and single-and multi-cell communications, with the designs aiming at minimizing the channel estimation error or maximizing achievable rates with reduced training overhead. Theoretical analysis and numerical performance results indicate significant reduction of training overhead over conventional techniques while achieving similar performance. The presented methods demonstrate the importance of exploiting fundamental channel properties and reveal important insights on the interplay/tradeoff between training overhead and performance that can serve as guidelines for the design of future massive MIMO communication systems.