**We
review the cases for which the 1D
stationary Schrödinger equation is
solved in terms of the general and
(multi-)confluent Heun functions.
We present the possible choices
for coordinate transformation that
provide energy-independent
potentials that are proportional
to an energy-independent
continuous parameter and have a
shape independent of that
parameter. In contrast to the
hypergeometric case, no Heun
potential can in general be
transformed into another one by
specifications of the involved
parameters.**

We show that there exist in total
29 independent Heun potentials.
There are eleven independent
potentials that admit the solution
in terms of the general Heun
functions, for nine independent
seven-parametric potentials the
solution is given in terms of the
single-confluent Heun functions,
there are three independent
double-confluent and five
independent bi-confluent Heun
potentials (the six-parametric
Lemieux-Bose potentials), and one
tri-confluent Heun potential (the
general five-parametric quartic
oscillator).

There are several independent
potentials that present distinct
generalizations of either a
hypergeometric or a confluent
hypergeometric classical
potential, some potentials possess
sub-cases of both hypergeometric
types, and others possess
particular conditionally
integrable ordinary or confluent
hypergeometric sub-potentials. We
present several examples of
explicit solutions for the latter
potentials.

We show that there exist other
exactly or conditionally
integrable sub-potentials the
solution for which is written in
terms of simpler special
functions. However, these are
solutions of different structure.
For instance, there are
sub-potentials for which each of
the two fundamental solutions of
the Schrödinger equation is
written in terms of irreducible
combinations of hypergeometric
functions. Several such potentials
are derived with the use of
deformed Heun equations. A
complementary approach is the
termination of the hypergeometric
series expansions of the solutions
of the Heun equations.