In this paper, we study the relationship between forbidden subgraphs and the existence of a matching. Let W be a set of connected graphs. each of which has three or more vertices. A graph G is said to be H-free if no graph in W is ail induced subgraph of G. We completely characterize the set H such that every connected H-free graph of sufficiently large even order has a perfect matching in the following cases. (1) Every graph in R is triangle-free. (2) H consists of two graphs (i.e. a pair of forbidden subgraphs). A matching M in a graph of odd order is said to be a near-perfect matching if every vertex of G but one is incident with an edge of M. We also characterize H such that every H-free graph of sufficiently large odd order has a near-perfect matching in the above cases. (C) 2005 Elsevier Inc. All rights reserved.963315324