Structural analysis is often viewed by the nonengineer as a mysterious and magical process leading to the Truth. Attempts by engineers to demystify the process have not always been successful, which has led many nonengineers interested in ancient structures to the conclusion that if one can simply find an amiable engineer competent with a computer all structural questions can be readily answered. In fact, much of the basic understanding of arch and vault behavior was developed long before computers became available. In any case, the computer provides answers only as accurate as the information entered. Moreover, different approaches can be applied to structural analysis, and there is some debate regarding which approach yields the most useful information. My goal in this chapter is to make yet another attempt to demystify the basics of structural analysis and to present the two major approaches most often applied to the analysis of historical structures, finite element modeling and thrust line analysis. Each approach makes different assumptions about the structure and asks different questions; therefore, the answers are not always comparable. The choice of approach depends in part on the question one is trying to answer. For the nonengineer interested in the structural behavior of historical buildings, understanding the fundamental differences provides some basis for evaluating the results. After discussing the historical development of arch analysis and the modern approaches to it, I present a series of case studies intended to demonstrate various ways in which structural analysis can be useful to the archaeologist examining ancient buildings.