This dissertation reports on a systematic investigation of student understanding of the concepts of time and space in special relativity. During the investigation we have identified persistent difficulties with the definitions of the position and time of an event and with the concept of a reference frame. Many students do not think of a reference frame as a system of observers that determine the same position and time for any event. Instead, they interpret statements of the frame-dependence of the time of an event to mean that observers at different locations receive signals from events at different times. When asked to describe measurement procedures for spatial quantities, students do not spontaneously apply the formalism of a reference frame, but instead tend to associate events with moving objects in a manner consistent with indiscriminate application of length contraction.  Traditional instruction in relativity appears to have little effect on these ideas, which are present among students from the introductory to the graduate level in physics. We have applied the results from this research to guide the design of instructional materials to address some of the specific difficulties that we identified.