A hexagonal system (HS) is a finite plane graph with no cut-vertices in which every interior region is a hexagonal unit cell. Assume that the vertices of an HS have been colored white and black. We let B(H) and W(H) denote the sets of black and white vertices, respectively, of the hexagonal system H. An edge-cut (EC) of an HS H is a collection of edges of H such that the subgraph H-EC obtained from H by deleting all edges in EC has more components than H. The authors prove the following necessary and sufficient condition for an HS to have a perfect matching. Let H be an HS such that \(| B(H)| =| W(H)|\). The H has a perfect matching if and only if for each edge-cut \(EC=\{e_ 1,...,e_ t\}\) satisfying the following three conditions, we have \(| B(G')| \geq | W(G')|\). (1) H-EC has exactly two components G' and G''. (2) The end vertex in G' of each \(e_ i\), \(i=1,...,t\), has the same color. (3) Edges \(e_ 1\) and \(e_ t\) lie on the boundary of H, and \(e_ i\) and \(e_{i+1}\) are edges of some hexagonal unit cell for every i, \(1\leq i\leq t-1\).