By integrating the storage and computing functions on the fundamental elements, computing in-memory (CIM) technology is widely considered as a novel computational paradigm that can break the bottleneck of Von Neumann architecture. Nonvolatile memory device is an appropriate hardware implementation approach of CIM, which possess significantly advantages, such as excellent scalability, low consumption, and versatility. In this paper, first we introduce the basic concept of CIM, including the technical background and technical characteristics. Then, we review the traditional and novel nonvolatile memory devices, flash and resistive random access memory (RRAM), used in non-volatile based computing in-memory (nvCIM) system. After that, we explain the operation modes of nvCIM: in-memory analog computing and in-memory digital computing. In addition, the applications of nvCIM are also discussed, including deep learning accelerator, neuromorphic computing, and stateful logic. Finally, we summarize the current research advances in nvCIM and provide an outlook on possible research directions in the future.