  A comprehensive review of the discrete quantum mechanics with the pure
imaginary shifts and the real shifts is presented in parallel with the
corresponding results in the ordinary quantum mechanics. The main subjects to
be covered are the factorised Hamiltonians, the general structure of the
solution spaces of the Schroedinger equation (Crum's theorem and its
modification), the shape invariance, the exact solvability in the Schroedinger
picture as well as in the Heisenberg picture, the creation/annihilation
operators and the dynamical symmetry algebras, the unified theory of exact and
quasi-exact solvability based on the sinusoidal coordinates, the infinite
families of new orthogonal (the exceptional) polynomials. Two new infinite
families of orthogonal polynomials, the X_\ell Meixner-Pollaczek and the X_\ell
Meixner polynomials are reported.
