  Let $K$ denote a field, and let $V$ denote a vector space over $K$ with
finite positive dimension. We consider a pair of linear transformations $A:V\to
V$ and $A^*:V\to V$ that satisfy the following two conditions:
  There exists a basis for $V$ with respect to which the matrix representing
$A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal.
  There exists a basis for $V$ with respect to which the matrix representing
$A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal.
  We call such a pair a Leonard pair on $V$. We give a correspondence between
Leonard pairs and a class of orthogonal polynomials. This class coincides with
the terminating branch of the Askey scheme and consists of the $q$-Racah,
$q$-Hahn, dual $q$-Hahn, $q$-Krawtchouk, dual $q$-Krawtchouk, quantum
$q$-Krawtchouk, affine $q$-Krawtchouk, Racah, Hahn, dual Hahn, Krawtchouk,
Bannai/Ito, and orphan polynomials. We describe the above correspondence in
detail. We show how, for the listed polynomials, the 3-term recurrence,
difference equation, Askey-Wilson duality, and orthogonality can be expressed
in a uniform and attractive manner using the corresponding Leonard pair. We
give some examples that indicate how Leonard pairs arise in representation
theory and algebraic combinatorics. We discuss a mild generalization of a
Leonard pair called a tridiagonal pair. At the end we list some open problems.
Throughout these notes our argument is elementary and uses only linear algebra.
No prior exposure to the topic is assumed.
