Techniques of medical image processing and analysis play a crucial role in many clinical scenarios, including in diagnosis and treatment planning. However, immense quantities of data and high complexity of the algorithms often used are computationally demanding. As a result, there now exists a wide range of techniques of medical image processing and analysis that require the application of high-performance computing solutions in order to reduce the required runtime. The main purpose of this review is to provide a comprehensive reference source of techniques of medical image processing and analysis that have been accelerated by high-performance computing solutions. With this in mind, the articles available in the Scopus and Web of Science electronic repositories were searched. Subsequently, the most relevant articles found were individually analyzed in order to identify: (a) the metrics used to evaluate computing performance, (b) the high-performance computing solution used, (c) the parallel design adopted, and (d) the task of medical image processing and analysis involved. Hence, the techniques of medical image processing and analysis found were identified, reviewed, and discussed, particularly in terms of computational performance. Consequently, the techniques reviewed herein present the progress made so far in reducing the computational runtime involved, and the difficulties and challenges that remain to be overcome.