  In the first part of the paper we derive integral curvature estimates for
complete gradient shrinking Ricci solitons. Our results and the recent work of
Lopez-Rio imply rigidity of gradient shrinking Ricci solitons with harmonic
Weyl tensor. In the second part of the paper we address the issue of existence
of harmonic functions on gradient shrinking K\"{a}hler and gradient steady
Ricci solitons and show that if the total energy of a harmonic function on such
a manifold is finite then the function is constant. Consequences to the
structure of the manifold at infinity are also discussed.
