Two of the most popular approaches to supervised learning are kernels and artificial neural networks (ANN).The ability to emulate complicated nonlinear behavior makes them powerful tools for function approximation.Randomization schemes, which have been successfully deployed to improve efficiency for many algorithms, havealso been developed for kernel methods and ANNs. These are random weight neural networks (RWNN) and their kernelanalogue, random features. We discuss the complicated origins of RWNNs and random features,along with the close connections between them. We summarize important theoretical results and future avenues of study.