Recent advances in deep learning have led to increased interest in solving high-efficiency end-to-end transmission problems using methods that employ the nonlinear property of neural networks. These techniques, we call neural joint source-channel coding (NeurJSCC), extract latent semantic features of the source signal across space and time, and design corresponding variable-length NeurJSCC approaches to transmit latent features over wireless communication channels. Rapid progress has led to numerous research papers, but a consolidation of the discovered knowledge has not yet emerged. In this article, we gather diverse ideas to categorize the expansive aspects on NeurJSCC as two paradigms, i.e., explicit and implicit NeurJSCC. We first focus on those two paradigms of NeurJSCC by identifying their common and different components in building end-to-end communication systems. We then focus on typical applications of NeurJSCC to various communication tasks. Our article highlights the improved quality, flexibility, and capability brought by NeurJSCC, and we also point out future directions.