High performance catalysts are crucial to generating clean fuels, reducing the impact of global warming, and providing solutions to environmental pollution. Improved processes for catalyst design and a better understanding of catalytic processes are key for improving the effectiveness and activities. HEAs typically have at least four principal elements, this atomic structure gives them unique properties that have applications and excellent performance in a variety of fields including catalysis. The complexity of HEAs makes challenge for computational researchers, providing promising opportunities for the application of machine learning. Recent advances in data science have great potential to accelerate catalyst research, particularly the rapid exploration of large materials chemistry spaces through machine learning. Here a comprehensive and critical review of machine learning techniques used in HEA catalysis research is provided. Sources of HEA catalyst data and current approaches to represent these materials by mathematical features are described, the most commonly used machine learning methods summarized, and the quality and utility of catalyst models evaluated. Illustrations of how machine learning models are applied to novel HEA catalysts discovery and used to reveal catalytic reaction mechanisms are provided.