 Two related slogans for structuralism in the philosophy of mathematics are that “mathematics is the general study of structures” and that, in pursuing such study, we can “abstract away from the nature of objects instantiating those structures”. (As such, structuralism stands in contrast with several other general views about mathematics, including: the traditional view that mathematics is the science of number and quantity; the view that it is an empty formalism used primarily for calculation; and the view that it is the study of a basic set-theoretic universe.) As the present survey aims to show, these slogans, while suggestive, are ambiguous and in need of clarification. Indeed, they have been interpreted in various different, even conflicting ways.