  The complete quantum theory of covariant closed strings is constructed in
detail. The action is defined by elementary vertices satisfying recursion
relations that give rise to Jacobi-like identities for an infinite chain of
string field products. The genus zero string field algebra is the homotopy Lie
algebra $L_\infty$, and the higher genus algebraic structure implies the
Batalin-Vilkovisky (BV) master equation. From these structures on the off-shell
state space, we show how to derive the $L_\infty$ algebra, and the BV equation
on physical states, recently constructed in d=2 string theory. The string
diagrams are surfaces with minimal area metrics, foliated by closed geodesics
of length $2\pi$. These metrics generalize quadratic differentials in that
foliation bands can cross. The string vertices are succinctly characterized;
they include the surfaces whose foliation bands are all of height smaller than
$2\pi$.
  --While this is not a review paper, an effort was made to give a fairly
complete and accessible account of the quantum closed string field theory.--
