  In this paper we introduce a notion of rational singularities associated to
pairs $(X, \ba^t)$ where $X$ is a variety, $\ba$ is an ideal sheaf and $t$ is a
nonnegative real number. We prove that most standard results about rational
singularities extend to this context. We also show that some results commonly
associated with log terminal pairs have analogs in this context, including
results related to inversion of adjunction. A positive characteristic analogue
of rational singularities of pairs is also defined and explored.
