  Given two smooth projective varieties X and Y over a field, we say that X
motivates Y if the (suitably defined) motive of Y is contained in the category
generated from X by taking sums, summands and products. This notion has
appeared implicitly in many places, but it seems useful to make it explicit.
  Some techniques are given for checking this condition, but in a nutshell it
involves building a correspondence which "dominates" Y. Among the more
interesting examples dealt with are moduli spaces. We show that in number of
cases moduli spaces of sheaves over curves or surfaces are motivated by underly
curve or surface. This allows us to check (or recheck) the (generalized) Hodge
and Lefschetz standard conjectures for some of these examples. The dominating
correspondence in these examples is built from the universal sheaf, and can, in
some instances, be realized as a kind of Fourier-Mukai transform.
