  Following the pioneering work of Duistermaat and Gr\"unbaum, we call a family
$\{p_n(x)\}_{n=0}^{\infty}$ of polynomials bispectral, if the polynomials are
simultaneously eigenfunctions of two commutative algebras of operators: one
consisting of difference operators acting on the degree index $n$, and another
one of operators acting on the variable $x$. The goal of the present paper is
to construct and parametrize bispectral extensions of the Askey-Wilson
polynomials, where the second algebra consists of $q$-difference operators. In
particular, we describe explicitly measures on the real line for which the
corresponding orthogonal polynomials satisfy (higher-order) $q$-difference
equations extending all known families of orthogonal polynomials satisfying
$q$-difference, difference or differential equations in $x$.
