With the upcoming trend of Big Data era, some new types of memory technologies have emerged as substitutes for the traditional Si-based semiconductor memory devices, which are encountering severe scaling down technical obstacles. In particular, the resistance random access memory (RRAM) and magnetic random access memory (MRAM) hold great promise for the in-memory computing, which are regarded as the optimal strategy and pathway to solve the von Neumann bottleneck by high-throughput in situ data processing. As far as the active materials in RRAM and MRAM are concerned, organic semiconducting materials have shown increasing application perspectives in memory devices due to their rich structural diversity and solution processability. With the introduction of metal elements into the backbone of molecules, some new properties and phenomena will emerge accordingly. Consequently, the RRAM and MRAM devices based on metal-containing organic compounds (including the small molecular metal complexes, metallopolymers, metal-organic frameworks (MOFs) and organic-inorganic-hybrid perovskites (OIHPs)) have been widely explored and attracted intense attention. In this review, we highlight the fundamentals of RRAM and MRAM, as well as the research progress of the applications of metal-containing organic compounds in both RRAM and MRAM. Finally, we discuss the challenges and future directions for the research of organic RRAM and MRAM.