A charm of biology as a scientific discipline is the diversity of life. Although this diversity can make laws of biology challenging to discover, several repeated patterns and general principles govern evolutionary diversification. Convergent evolution, the independent evolution of similar phenotypes, has been at the heart of one approach to understand generality in the evolutionary process. Yet understanding when and why organismal traits and strategies repeatedly evolve has been a central challenge. These issues were the focus of the American Society of Naturalists Vice Presidential Symposium in 2016 and are the subject of this collection of articles. Although naturalists have long made inferences about convergent evolution and its importance, there has been confusion in the interpretation of the pattern of convergence. Does convergence primarily indicate adaptation or constraint? How often should convergence be expected? Are there general principles that would allow us to predict where and when and by what mechanisms convergent evolution should occur? What role does natural history play in advancing our understanding of general evolutionary principles? In this introductory article, I address these questions, review several generalizations about convergent evolution that have emerged over the past 15 years, and present a framework for advancing the study and interpretation of convergence. Perhaps the most important emerging conclusion is that the genetic mechanisms of convergent evolution are phylogenetically conserved; that is, more closely related species tend to share the same genetic basis of traits, even when independently evolved. Finally, I highlight how the articles in this special issue further develop concepts, methodologies, and case studies at the frontier of our understanding of the causes and consequences of convergent evolution.