The principle of equivalence was the first vital clue to Einstein in his extension of special relativity to general relativity, the modern theory of gravitation. In this paper we investigate in some detail students' understanding of this principle in a variety of contexts, when they are undergoing an introductory course on general relativity. The tool of assessment is a diagnostic questionnaire administered to a group of 30 senior undergraduates going through the course. The analysis of responses reveals a number of significant conceptual vulnerabilities in several aspects of the topic. The most important among these is related to the notion of an inertial frame that follows naturally from the principle of equivalence. (Contains 18 tables, 6 figures, and 5 footnotes.)