Materials science is undergoing a revolution, generating valuable new materials such as flexible solar panels, biomaterials and printable tissues, new catalysts, polymers, and porous materials with unprecedented properties. However, the number of potentially accessible materials is immense. Artificial evolutionary methods such as genetic algorithms, which explore large, complex search spaces very efficiently, can be applied to the identification and optimization of novel materials more rapidly than by physical experiments alone. Machine learning models can augment experimental measurements of materials fitness to accelerate identification of useful and novel materials in vast materials composition or property spaces. This review discusses the problems of large materials spaces, the types of evolutionary algorithms employed to identify or optimize materials, and how materials can be represented mathematically as genomes, describes fitness landscapes and mutation operators commonly employed in materials evolution, and provides a comprehensive summary of published research on the use of evolutionary methods to generate new catalysts, phosphors, and a range of other materials. The review identifies the potential for evolutionary methods to revolutionize a wide range of manufacturing, medical, and materials based industries.