 We had two main goals in this essay. The first was to introduce our readers to the general discussion of structuralism in contemporary philosophy of mathematics.  Second, we tried to provide a richer, more inclusive taxonomy for that discussion, including making evident that a much larger variety of structuralist positions has played a role in it than is usually acknowledged.  To be sure, a certain amount of variety has been recognized before, as reflected in the distinction between eliminative and non-eliminative positions, with Shapiro’s ante rem structuralism and Hellman’s modal structuralism as paradigms; and categorical structuralism has been acknowledged as a third main alternative. But the range of positions at play is much greater than that, especially in the recent literature.  “Mathematical structuralism” is not the name of a single position, but of a multifaceted family of them.