 There are two main reasons for promoting our more inclusive taxonomy. The first is that, while numerous versions of structuralism for mathematics have been proposed, several of them have not received much attention yet, partly because their relationships to Shapiro’s, Hellman’s, and category-theoretic positions have remained unclear.  Along the way, we tried to build some corresponding bridges. The second reason is that several important variants played a role already before the 1960s, i.e., before the publications by Benacerraf and Putnam that are often seen as the starting point for debates about structuralism in philosophy. This concerns both methodological and metaphysical structuralism, a basic distinction we highlighted; but we only hinted at the relevant “pre-history” of our topic in the present survey (We refer the reader to Reck &amp; Schiemer forthcoming for more on it.)