  The Heun-Askey-Wilson algebra is introduced through generators
$\{\boX,\boW\}$ and relations. These relations can be understood as an
extension of the usual Askey-Wilson ones. A central element is given, and a
canonical form of the Heun-Askey-Wilson algebra is presented. A homomorphism
from the Heun-Askey-Wilson algebra to the Askey-Wilson one is identified. On
the vector space of the polynomials in the variable $x=z+z^{-1}$, the Heun
operator of Askey-Wilson type realizing $\boW$ can be characterized as the most
general second order $q$-difference operator in the variable $z$ that maps
polynomials of degree $n$ in $x=z+z^{-1}$ into polynomials of degree $n+1$.
