We consider a deep learning (DL) based approach to optimize channel state information (CSI) reporting in massive multiple-input and multiple-output (MIMO) systems. For CSI compression, existing methods use frequency-domain (FD) and spatial-domain (SD) correlation, whereas correlation also exists in the time-domain (TD) under fading conditions. Hence, we propose a DL-based three-dimensional compression (DL-3DC) approach to improve CSI reporting accuracy by using correlation in FD, SD and TD. In addition, to reduce the feedback overhead, we propose two DL-based CSI prediction methods: eigenvector based and MIMO channel based. We then integrate CSI prediction with DL-3DC at the UE and propose a joint CSI compression and prediction (JCCP) scheme to improve the CSI accuracy and reporting overhead trade-off. Although, UE-side JCCP is efficient when the future CSI application time instance is known in advance. To support prediction when this is unknown, we propose BS-side JCCP to predict the CSI at the BS after it is received. Through simulations, we show that DL-3DC improves the CSI reporting accuracy by up to 13.8% compared to the latest CSI reporting mechanism in New Radio (NR). Further, we show that JCCP significantly reduces the reporting overhead by up to 87.5% compared to NR CSI reporting.