Much of mathematics involves proving structure theorems for classes of objects — the idea being to give a complete list of such objects up to isomorphism. For example, the classification of vector spaces over a field F says that any vector space is isomorphic to a direct sum of copies of F. A similar theorem (of Steinitz) says that every algebraically closed field is isomorphic to the algebraic closure of the field of rational functions in some (potentially infinite) number of variables over either the rationals or the integers mod a prime. Classification theory — a topic within model theory — uses first-order logic to try to formalize the question “which mathematical theories admit structure theorems?” Attempts to answer this question were at the forefront of early modern model theory, and the main ideas continue to guide the direction of the field today. In this talk, I will give a historical overview of classification theory, focusing on the most basic examples and questions. Note that no knowledge of logic or model theory is assumed, and the talk could equally well have been titled “what is model theory”.