Many performance gains achieved by massive multiple-input and multiple-output depend on the accuracy of the downlink channel state information (CSI) at the transmitter (base station), which is usually obtained by estimating at the receiver (user terminal) and feeding back to the transmitter. The overhead of CSI feedback occupies substantial uplink bandwidth resources, especially when the number of the transmit antennas is large. Deep learning (DL)-based CSI feedback refers to CSI compression and reconstruction by a DL-based autoencoder and can greatly reduce feedback overhead. In this paper, a comprehensive overview of state-of-the-art research on this topic is provided, beginning with basic DL concepts widely used in CSI feedback and then categorizing and describing some existing DL-based feedback works. The focus is on novel neural network architectures and utilization of communication expert knowledge to improve CSI feedback accuracy. Works on bit-level CSI feedback and joint design of CSI feedback with other communication modules are also introduced, and some practical issues, including training dataset collection, online training, complexity, generalization, and standardization effect, are discussed. At the end of the paper, some challenges and potential research directions associated with DL-based CSI feedback in future wireless communication systems are identified.