  We explain how the moments of the (weight function of the) Askey Wilson
polynomials are related to the enumeration of the staircase tableaux introduced
by the first and fourth authors. This gives us a direct combinatorial formula
for these moments. Then we use techniques developed by Ismail and the third
author to give explicit formulae for these moments and for the enumeration of
staircase tableaux. Finally we study the enumeration of staircase tableaux at
various specializations of the parameterizations; for example, we obtain the
Catalan numbers, Fibonacci numbers, Eulerian numbers, the number of
permutations, and the number of matchings.
