  We introduce the concept of boundary degeneracy of topologically ordered
states on a compact orientable spatial manifold with boundaries, and emphasize
that the boundary degeneracy provides richer information than the bulk
degeneracy. Beyond the bulk-edge correspondence, we find the ground state
degeneracy of the fully gapped edge modes depends on boundary gapping
conditions. By associating different types of boundary gapping conditions as
different ways of particle or quasiparticle condensations on the boundary, we
develop an analytic theory of gapped boundaries. By Chern-Simons theory, this
allows us to derive the ground state degeneracy formula in terms of boundary
gapping conditions, which encodes more than the fusion algebra of
fractionalized quasiparticles. We apply our theory to Kitaev's toric code and
Levin-Wen string-net models. We predict that the $Z_2$ toric code and $Z_2$
double-semion model (more generally, the $Z_k$ gauge theory and the $U(1)_k
\times U(1)_{-k}$ non-chiral fractional quantum Hall state at even integer $k$)
can be numerically and experimentally distinguished, by measuring their
boundary degeneracy on an annulus or a cylinder.
