The book generalizes the well-known regularity of ring elements to regularity of homomorphisms in module categories, and further to regularity of morphisms in any category. Regular homomorphisms are characterized in terms of decompositions of domain and codomain, and numerous other results are presented. While the theory is well developed in module categories many questions remain about generalizations and extensions to other categories. The book only requires the knowledge of a basic course in modern algebra. It is written clearly, with great detail, and is accessible to students and researchers alike.