To understand the meaning of evolutionary equilibria, it is necessary to comprehend the ramifications of the evolutionary model. For instance, a full appreciation of Axelrod's The Evolution of Cooperation requires that we identify assumptions under which conditionally cooperative strategies, like Tit For Tat, are and are not evolutionarily stable. And more generally, when does stability fail? To resolve these questions we re-examine the very foundations of the evolutionary model. The results of this paper can be analytically separated into three parts. The first part is conceptual: it identifies the evolutionary model's assumptions and shows how different assumptions imply different types of evolutionary stability. The second part is deductive: it establishes necessary and sufficient conditions for the types of evolutionary stability identified in the first part, and demonstrates in which games these kinds of stability can (and cannot) be attained. The third and final part is applied: it relates the general findings (which are independent of the specific payoffs of any particular evolutionary game) to the issue of the evolutionary stability of cooperation. Results on cooperation appear throughout the paper as they both exemplify and motivate the general results. These results essentially explain when cooperation is and is not stable, and why, thus shedding new light on the meaning and applicability of Axelrod's widely known claims.