In this article I report on a study of the cognitive tools that research mathematicians employ when developing deep understandings of abstract mathematical definitions. I arrived at several conclusions about this process: Examples play a predominant role in understanding definitions. Equivalent reformulations of definitions enrich understanding. Evoked conflicts and their resolutions result in improved understanding. The primary role of definitions in mathematics is in proving theorems. And there are several stages in developing understandings of mathematical definitions (Manin, 2007; Tall & Vinner, 1981; Thurston, 1994). I also include some suggestions for pedagogy that are found in the data.