  We prove short time existence and uniqueness of the Laplacian flow starting
at an arbitrary closed $G_2$-structure. We establish long time existence and
convergence of the Laplacian flow starting near a torsion-free $G_2$-structure.
We analyze the limit map of the Laplacian flow in relation to the moduli space
of torsion-free $G_2$-structures. We also present a number of results which
constitute a fairly complete algebraic and analytic basis for studying the
Laplacian flow.
