  A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used
to define a unitary quantum system, if one modifies the inner product of the
Hilbert space properly. We give a comprehensive and essentially self-contained
review of the basic ideas and techniques responsible for the recent
developments in this subject. We provide a critical assessment of the role of
the geometry of the Hilbert space in conventional quantum mechanics to reveal
the basic physical principle motivating our study. We then offer a survey of
the necessary mathematical tools and elaborate on a number of relevant issues
of fundamental importance. In particular, we discuss the role of the antilinear
symmetries such as PT, the true meaning and significance of the charge
operators C and the CPT-inner products, the nature of the physical observables,
the equivalent description of such models using ordinary Hermitian quantum
mechanics, the pertaining duality between local-non-Hermitian versus
nonlocal-Hermitian descriptions of their dynamics, the corresponding classical
systems, the pseudo-Hermitian canonical quantization scheme, various methods of
calculating the (pseudo-) metric operators, subtleties of dealing with
time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation
of the theory, and the structure of the state space and its ramifications for
the quantum Brachistochrone problem. We also explore some concrete physical
applications of the abstract concepts and tools that have been developed in the
course of this investigation. These include applications in nuclear physics,
condensed matter physics, relativistic quantum mechanics and quantum field
theory, quantum cosmology, electromagnetic wave propagation, open quantum
systems, magnetohydrodynamics, quantum chaos, and biophysics.
