For an increasing number of preclinical samples, both detailed molecular profiles and their responses to various drugs are becoming available. Efforts to understand, and predict, drug responses in a data-driven manner have led to a proliferation of machine learning (ML) methods, with the longer term ambition of predicting clinical drug responses. Here, we provide a uniquely wide and deep systematic review of the rapidly evolving literature on monotherapy drug response prediction, with a systematic characterization and classification that comprises more than 70 ML methods in 13 subclasses, their input and output data types, modes of evaluation, and code and software availability. ML experts are provided with a fundamental understanding of the biological problem, and how ML methods are configured for it. Biologists and biomedical researchers are introduced to the basic principles of applicable ML methods, and their application to the problem of drug response prediction. We also provide systematic overviews of commonly used data sources used for training and evaluation methods.