In recent years it has been realized that Hermiticity of the Hamiltonian is not a necessary condition for the reality of its eigenvalues. In many cases it is sufficient for the Hamiltonian to have an unbroken PT symmetry, but the natural Hilbert-space metric of such theories is not positive definite, which raises severe problems as to their physical interpretation. However, it turns out to be possible to construct another,dynamically-determined, metric eta=exp(iQ) that is indeed positive definite. For the most part Q can only be constructed perturbatively,but there are some cases where it is known exactly. With the aid of Q one can construct an equivalent Hermitian Hamiltonian, which, however, may be rather complicated and non-local.