In this paper, we discuss the convergence of recent advances in deep neural networks (DNNs) with the design of robotic mechanisms, which entails the conceptualization of the design problem as a learning problem from the space of design specifications to a parameterization of the space of mechanisms. We identify three key inter-related problems that are at the forefront of using the versatility of DNNs in solving mechanism design problems. The first problem is that of representation of mechanisms and their design specifications, where the representation challenges arise primarily from the non-Euclidean nature of the data. The second problem is that of developing a mapping from the space of design specifications to the mechanisms where, ideally, we would like to synthesize both type and dimensions of the mechanism for a wide variety of design specifications including path synthesis, motion synthesis, constraints on pivot locations, etc. The third problem is that of designing the neural network architecture for end-to-end training and generation of multiple candidate mechanisms for a given design specification. We also present a brief overview of the state-of-the-art on each of these problems and identify questions of potential interest to the research community.