Let $G=(V,E)$ be a finite simple undirected graph. The convex hull of incidence vectors of matchings in $G$ is the set of all $x \in \mathbb R^E$ satisfying the degree inequalities $\sum_{e \in \delta(v)}x_e \le 1$ for all $v \in V$, the non-negativity inequalities $x_e \ge 0$ for all $e \in E$, and all odd-set inequalities $\sum_{e \in E(S)}x_e \le (|S|-1)/2$ for odd $S \subset V$ with $|S|\ge 3$.