  We develop the topological band theory for systems described by non-Hermitian
Hamiltonians, whose energy spectra are generally complex. After generalizing
the notion of gapped band structures to the non-Hermitian case, we classify
"gapped" bands in one and two dimensions by explicitly finding their
topological invariants. We find nontrivial generalizations of the Chern number
in two dimensions, and a new classification in one dimension, whose topology is
determined by the energy dispersion rather than the energy eigenstates. We then
study the bulk-edge correspondence and the topological phase transition in two
dimensions. Different from the Hermitian case, the transition generically
involves an extended intermediate phase with complex-energy band degeneracies
at isolated "exceptional points" in momentum space. We also systematically
classify all types of band degeneracies.
