In the 1970s, structural Ramsey theory emerged as a new branch of combinatorics. This development came with the isolation of the concepts of the A-Ramsey property and Ramsey class. Following the influential Nešetřil–Rödl theorem, several Ramsey classes have been identified. In the 1980s, Nešetřil, inspired by a seminar of Lachlan, discovered a crucial connection between Ramsey classes and Fraïssé classes, and, in his 1989 paper, connected the classification programme of homogeneous structures to structural Ramsey theory. In 2005, Kechris, Pestov, and Todorčević revitalized the field by connecting Ramsey classes to topological dynamics. This breakthrough motivated Nešetřil to propose a program for classifying Ramsey classes. We review the progress made on this program in the past two decades, list open problems, and discuss recent extensions to new areas, namely the extension property for partial automorphisms (EPPA), and big Ramsey structures.