Power converters and motor drives are playing a significant role in the transition towards sustainable energy systems and transportation electrification. In this context, rich diversity of new power converters and motor drive products are developed and commissioned by the industry every year. However, to achieve efficient, reliable and stable operation of power converter and drive systems, there are challenges in condition monitoring, fault diagnosis, lifecycle estimation, stability evaluation and control. Online learning is an emerging technology that can serve as a powerful remedy to these challenges. This paper aims to provide a systematic study of algorithms, implementations, and applications of online learning for control and diagnostics in the area of power converters and drives. First, online learning problems are formulated for condition monitoring, fault detection, online stability assessment, model predictive control for power converter and drive applications. Next, guidelines are provided about how to develop online learning models and algorithms for these applications. Practical case studies are presented with experimental demonstrations. Finally, challenges and future opportunities are discussed about online learning for power converter and drive applications.