This paper presents a state-of-the-art review of different approaches for Neural Architecture Search targeting resource-constrained devices such as microcontrollers, as well as the implementations of on-device learning techniques for them. Approaches such as MCUNet have been able to drive the design of tiny neural architectures with low memory and computational requirements which can be deployed effectively on microcontrollers. Regarding on-device learning, there are various solutions that have addressed concept drift and have coped with the accuracy drop in real-time data depending on the task targeted, and these rely on a variety of learning methods. For computer vision, MCUNetV3 uses backpropagation and represents a state-of-the-art solution. The Restricted Coulomb Energy Neural Network is a promising method for learning with an extremely low memory footprint and computational complexity, which should be considered for future investigations.