To achieve the potential performance gain of massive multiple-input multiple-output (MIMO) systems, base stations (BS) require downlink channel state information (CSI) fed back by users to execute beamforming design, especially in the frequency division duplex (FDD) systems. However, due to the enormous number of antennas in massive MIMO systems, the feedback overhead of downlink CSI acquisition is extremely large. To address this issue, deep learning (DL) techniques have been introduced to develop high-accuracy feedback strategies under limited backhaul constraints. In this paper, we provide an overview of DL-based CSI compression and feedback approaches in massive MIMO systems. Specifically, we introduce the conventional CSI compression and feedback schemes and the existing problems. Besides, we elaborate on various DL techniques employed in CSI compression from the perspective of network architecture and analyze the advantages of different techniques. We also enumerate the applications of DL-based methods for solving practical challenges in CSI compression and feedback. In addition, we brief the remaining issues in deep CSI compression and indicate potential directions in future wireless networks.