Federated learning is an emerging technology that enables the decentralised training of machine learning-based methods for medical image analysis across multiple sites while ensuring privacy. This review paper thoroughly examines federated learning research applied to medical image analysis, outlining technical contributions. We followed the guidelines of Okali and Schabram, a review methodology, to produce a comprehensive summary and discussion of the literature in information systems. Searches were conducted at leading indexing platforms: PubMed, IEEE Xplore, Scopus, ACM, and Web of Science. We found a total of 433 papers and selected 118 of them for further examination. The findings highlighted research on applying federated learning to neural network methods in cardiology, dermatology, gastroenterology, neurology, oncology, respiratory medicine, and urology. The main challenges reported were the ability of machine learning models to adapt effectively to real-world datasets and privacy preservation. We outlined two strategies to address these challenges: non-independent and identically distributed data and privacy-enhancing methods. This review paper offers a reference overview for those already working in the field and an introduction to those new to the topic.