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This book provides a comprehensive study of so-called mathematical explanations of physical phenomena (MEPPs). It surveys the major philosophical views on this issue, provides the necessary background for understanding these debates, and addresses the main philosophical questions that arise from the existence of these explanations in science. The author defends a structuralist interpretation of MEPPs, and argues that, because the role of mathematics in these explanations is ultimately representational (in what he calls 'optimal representations'), MEPPs do not support mathematical realism. And despite these explanations being non-causal, they do not suffer the traditional problems of asymmetry and relevance that Hempel's DN model did. The book also introduces a novel case study, that of the Buckminsterfullerene (Buckyballs). This book is of interest to advanced undergraduate and graduate students interested in some of the fundamental philosophical problems in the applicability of mathematics in science, as well as advanced scholars interested in these issues.