We consider the Rankin-Selberg L-functions associated with a fixed modular
form of full level and holomorphic cuspidal newforms of large even weight,
fixed level and fixed primitive nebentypus. We compute the second moment of
this family in fairly general ranges, and obtain an asymptotic formula with a
power saving error term. A special case treats the fourth moment of L-functions
associated with holomorphic cusp forms.