In the era of widespread edge computing, energy conservation modes like complete power shutdown are crucial for battery-powered devices, but they risk data loss in volatile memory. Energy autonomous systems, relying on ambient energy, face operational challenges due to power losses. Recent advancements in emerging nonvolatile memories (NVMs) like FRAM, RRAM, MRAM, and PCM offer mature solutions to sustain work progress with minimal energy overhead during outages. This paper thoroughly reviews utilizing emerging NVMs in microcontroller units (MCUs), comparing their key attributes to describe unique benefits and potential applications. Furthermore, we discuss the intricate details of NVM circuit design and NVM-driven compute-in-memory (CIM) architectures. In summary, integrating emerging NVMs into MCUs showcases promising prospects for next-generation applications such as Internet of Things and neural networks.