This thesis is concerned with relative homological algebra and exact model categories. The word “relative” indicates that a certain choice of short exact sequences has been made. Then one considers (relative) approximations of objects in a category (all categories are additive in this thesis), with respect to the chosen class of short exact sequences, in a way analogous to the fibrant/cofibrant approximations of objects in model categories. There are two examples of such relative homological theories which are of interest in this thesis. One example comes from commutative algebra in the study of maximal Cohen–Macaulay approximations and its generalizations in Gorenstein homological algebra. Another example comes from the theory of purity in finitely accessible additive categories.