Machine Learning (ML) and Artificial Intelligence are omnipresent topics nowadays. Especially since small personal devices, such as smartphones, are capable of running computation intensive tasks on their own, they can improve our lives in many different ways, e.g. computer vision or speech detection. Deploying neural network models on tiny and therefore performance, memory and power-constrained devices, such as microcontrollers, is currently an important research topic. For mobile and other embedded target devices, Neural Architecture Search, or NAS for short, is a novel technique to find appropriate network models without requiring manual error-prone adjustments by engineers, which allows to automate the workflow from the definition of a real work problem to a deployable solution. While there are a lot of comprehensive surveys on NAS methods in general, none of them considers the application of those ideas on edge devices like microcontrollers (MCUs). This work provides an overview of the recent developments and compares different approaches, which appeared over the last few years, against one another and with well-known reference applications intended for the usage on mobile phones. In the end some opportunities for future work on this topic are discussed.