Machine learning in medical imaging often faces a fundamental dilemma, namely the small sample size problem. Many recent studies suggest using multi-domain data pooled from different acquisition sites/datasets to improve statistical power. However, medical images from different sites cannot be easily shared to build large datasets for model training due to privacy protection reasons. As a promising solution, federated learning, which enables collaborative training of machine learning models based on data from different sites without cross-site data sharing, has attracted considerable attention recently. In this paper, we conduct a comprehensive survey of the recent development of federated learning methods in medical image analysis. We first introduce the background and motivation of federated learning for dealing with privacy protection and collaborative learning issues in medical imaging. We then present a comprehensive review of recent advances in federated learning methods for medical image analysis. Specifically, existing methods are categorized based on three critical aspects of a federated learning system, including client end, server end, and communication techniques. In each category, we summarize the existing federated learning methods according to specific research problems in medical image analysis and also provide insights into the motivations of different approaches. In addition, we provide a review of existing benchmark medical imaging datasets and software platforms for current federated learning research. We also conduct an experimental study to empirically evaluate typical federated learning methods for medical image analysis. This survey can help to better understand the current research status, challenges and potential research opportunities in this promising research field.