We discuss the state-of-the-art on fault diagnosis for analog circuits with a focus on techniques that leverage machine learning. For a chip that has failed either in post-manufacturing testing or in the field of operation, fault diagnosis is launched to identify the root-cause of failure at sub-block level and transistor-level. In this context, machine learning can be used to build a smart system that predicts the fault that has occurred from diagnostic measurements extracted on the chip. We discuss the different elements of a diagnosis flow for analog circuits, including fault modeling, fault simulation, diagnostic measurement extraction and selection, and the machine learning algorithms that compose the prediction system. We also demonstrate a machine learning-based diagnosis flow on an industrial case study.