The topic of this chapter has become standard in modern treatments of differential geometry. The very words of the title have even been incorporated into part of a common cliche: Gauge theory is a connection on a principal bundle. We will come back to this relation between physics and geometry in Chapter 14 But just on the geometry side there has been an impressive amount of results, only a fraction of which we will be able to deal with here. Sometimes we speak of the need to translate geometric terminology into physics terminology. And vice versa. Curiously, there is also a need to translate geometrical terminology developed in one context into geometrical terminology from another context. And that is especially true for this topic. In this regard, the books [4] by Choquet-Bruhat and co-authors and [46] by Spivak are quite helpful references. Also quite readable is Darling’s text [6]. In an effort to keep this chapter as efficient as practically possible, we have not presented all the equivalent or closely related ways of approaching this central topic.