anindependentsetisasubsetofverticesofanundirectedgraph,inwhichnotwoverticesareadjacent. A clique is a subset of vertices of an undirected graph, in which every two distinct vertices are adjacent. For a graph H, we say that G is H-free if G has no induced subgraph isomorphic to H. In ????, Erd?s and Szekers proved that every graph on n vertices has either a clique or an independent set of size at least? ?log?(n). In ????, Erd?s proved that, for n sufficiently large there exists a graph on n vertices which has neither clique nor independent set of size ?log?(n). Having these results, while studying Ramsey-type problems in ????, Erd?s and Hajnal in ????, claimed that the situation is dramatically different for a fixed H-free graph. They states their conjecture as follows. Erd?s-Hajnal Conjecture. For every simple graph H, there exists a constant ?(H) gt; ? such that every H-free graph G has either a clique or an independent set of size at least |V (G)|?(H). In this thesis, we survey almost all known results relating this conjecture up to the tine of preparing this thesis. Erd?s and Hajnal proved for every graph H, there exists a constant c(H) gt; ? so that every H-free graph G has either a clique or an independent set of size at least ec(H)? log|V (G)|. We say, the graph H has Erd?s-Hajnal property if it satisfies in Erd?s-Hajnal Conjecture. It is clear that if H has Erd?s-Hajnal property, then¯H has this property, too. It was proved that every graph on at most ? vertices, except P?and C?has Erd?s-Hajnal property. The conjecture has remained open for P?and C?. For |V (H)| = ?, we have ?(H) = ?. For |V (H)| = ? and H = P?, we have ?(H) =? ?.