  The \emph{matching preclusion number} of a graph $G$, denoted by $\mpo(G)$,
is the minimum number of edges whose deletion results in a graph that has
neither perfect matchings nor almost-perfect matchings. In this paper, we first
give some sharp upper and lower bounds of matching preclusion number. Next,
graphs with large and small matching preclusion number are characterized,
respectively. In the end, we investigate some extremal problems and the
Nordhaus-Gaddum-type relations on matching preclusion number.
