  We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian
with a real spectrum is pseudo-Hermitian. We point out that all the
PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the
class of pseudo-Hermitian Hamiltonians, and argue that the basic structure
responsible for the particular spectral properties of these Hamiltonians is
their pseudo-Hermiticity. We explore the basic properties of general
pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics,
and study some concrete examples, namely the Hamiltonian of the two-component
Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar
field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
