We consider the problem of downlink training and channel estimation in frequency division duplex (FDD) massive MIMO systems, where the base station (BS) equipped with a large number of antennas serves a number of single-antenna users simultaneously. To obtain the channel state information (CSI) at the BS in FDD systems, the downlink channel has to be estimated by users via downlink training and then fed back to the BS. For FDD large-scale MIMO systems, the overhead for downlink training and CSI uplink feedback could be prohibitively high, which presents a significant challenge. In this paper, we study the behavior of the minimum mean-squared error (MMSE) estimator when the channel covariance matrix has a low-rank or an approximate low-rank structure. Our theoretical analysis reveals that the amount of training overhead can be substantially reduced by exploiting the low-rank property of the channel covariance matrix. In particular, we show that the MMSE estimator is able to achieve exact channel recovery in the asymptotic low-noise regime, provided that the number of pilot symbols in time is no less than the rank of the channel covariance matrix. We also present an optimal pilot design for the single-user case, and an asymptotic optimal pilot design for the multi-user scenario. Lastly, we develop a simple model-based scheme to estimate the channel covariance matrix, based on which the MMSE estimator can be employed to estimate the channel. The proposed scheme does not need any additional training overhead. Simulation results are provided to verify our theoretical results and illustrate the effectiveness of the proposed estimated covariance-assisted MMSE estimator.