  Quasi-Exactly Solvable Schr\"odinger Equations occupy an intermediate place
between exactly-solvable (e.g. the harmonic oscillator and Coulomb problems
etc) and non-solvable ones. Their major property is an explicit knowledge of
several eigenstates while the remaining ones are unknown. Many of these
problems are of the anharmonic oscillator type with a special type of
anharmonicity. The Hamiltonians of quasi-exactly-solvable problems are
characterized by the existence of a hidden algebraic structure but do not have
any hidden symmetry properties. In particular, all known one-dimensional
(quasi)-exactly-solvable problems possess a hidden $\mathfrak{sl}(2,\bf{R})-$
Lie algebra. They are equivalent to the $\mathfrak{sl}(2,\bf{R})$ Euler-Arnold
quantum top in a constant magnetic field. Quasi-Exactly Solvable problems are
highly non-trivial, they shed light on delicate analytic properties of the
Schr\"odinger Equations in coupling constant. The Lie-algebraic formalism
allows us to make a link between the Schr\"odinger Equations and
finite-difference equations on uniform and/or exponential lattices, it implies
that the spectra is preserved. This link takes the form of quantum canonical
transformation. The corresponding isospectral spectral problems for
finite-difference operators are described. The underlying Fock space formalism
giving rise to this correspondence is uncovered. For a quite general class of
perturbations of unperturbed problems with the hidden Lie algebra property we
can construct an algebraic perturbation theory, where the wavefunction
corrections are of polynomial nature, thus, can be found by algebraic means.
