AbstractMany graphs which are encountered in the study of graph theory are characterized by a type of configuration or subgraph they possess. However, there are occasions when such graphs are more easily defined or described by the kind of subgraphs they are not permitted to contain. For example, a tree can be defined as a connected graph which contains no cycles, and Kuratowski [22] characterized planar graphs as those graphs which fail to contain subgraphs homeomorphic from the complete graphK5 or the complete bipartite graphK3,3.The purpose of this article is to study, in a unified manner, several classes of graphs, which can be defined in terms of the kinds of subgraphs they do not contain, and to investigate related concepts. In the process of doing this, we show that many “apparently unrelated” results in the literature of graph theory are closely related. Several unsolved problems and conjectures are also presented.