This book gives a detailed account of the analytic foundations of gauge theory -- Uhlenbeck's compactness theorems for general connections and for Yang-Mills connections. It intends to guide graduate students into the analysis of Yang-Mills theory as well as to serve as a reference for researchers in the field. The book is largely self-contained. It contains a number of appendices (e.g. on Sobolev spaces of maps between manifolds) and an introductory part covering the $L^p$-regularity theory for the inhomogenous Neumann problem. The two main parts contain the full proofs of Uhlenbeck's weak and strong compactness theorems on closed manifolds as well as their generalizations to manifolds with boundary and noncompact manifolds. These parts include a number of useful analytic tools such as general patching constructions and local slice theorems.