Knowledge of information about the propagation channel in which a wireless system operates enables better, more efficient approaches for signal transmissions. Therefore, channel state information (CSI) plays a pivotal role in the system performance. The importance of CSI is in fact growing in the upcoming 5G and beyond systems, e.g., for the implementation of massive multiple-input multiple-output (MIMO). However, the acquisition of timely and accurate CSI has long been considered as a major issue, and becomes increasingly challenging due to the need for obtaining CSI of many antenna elements in massive MIMO systems. To cope with this challenge, existing works mainly focus on exploiting linear structures of CSI, such as CSI correlations in the spatial domain, to achieve dimensionality reduction. In this article, we first systematically review the state-of-the-art on CSI structure exploitation; then extend to seek for deeper structures that enable remote CSI inference wherein a data-driven deep neural network (DNN) approach is necessary due to model inadequacy. We develop specific DNN designs suitable for CSI data. Case studies are provided to demonstrate great potential in this direction for future performance enhancement.