Perfect graphs constitute a well-studied graph class with a rich structure. This is reflected by characterizations of perfect graphs with respect to such different concepts as coloring properties, forbidden subgraphs, the integrality of certain polyhedra, or splitting graph entropies. In addition, several otherwise hard combinatorial optimization problems can be solved for perfect graphs in polynomial time. Thus, perfect graphs play a role in such various mathematical disciplines as graph theory, information theory, combinatorial optimization, integer and semidefinite programming, polyhedral and convexity theory, thereby linking those disciplines in a truly unexpected way. Unfortunately, most graphs are imperfect and do not have such nice properties. It is, therefore, natural to ask which imperfect graphs are close to perfection in some sense and how to measure that. A canonical way is to relax several concepts which characterize perfect graphs and investigate the corresponding superclasses of perfect graphs. Perfect graphs are exceptional with respect to all the studied concepts; we address the question which graphs in the considered superclasses are `almost' perfect in several respects, and which are close to perfection w.r.t. one concept only, but not w.r.t. the others. In addition, we are also interested in linking the concepts that generalize the notion of perfection in different ways, e.g., in polyhedral terms, by means of splitting graph entropies, or w.r.t. more general coloring concepts. As conclusion, it will turn out that the imperfection ratio is an appropriate measure for imperfection in all these respects.