 So far we have traced the development of structuralism in the philosophy of mathematics from Benacerraf and Putnam, in the 1960s, to Resnik, Shapiro, Hellman, Chihara, and Parsons, in the 1980s–90s. During the last 20 years, a number of further philosophers have started to address this topic. We now turn to the corresponding discussions, starting with certain epistemological and metaphysical challenges to structuralism. Some of these concern non-eliminative structuralism alone, especially Shapiro’s position. (This again reflects the prominence of Shapiro’s position, to the point that it is often identified, misleadingly, with “philosophical structuralism”.) Others are broader, including comparing the basic commitments of various forms of structuralism. In what follows, we will not try to be comprehensive, but provide a number of illustrative examples.