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We generalize the notions of F-regular and F-pure rings to pairs $(R,\a^t)$of rings $R$ and ideals $\a \subset R$ with real exponent $t > 0$, andinvestigate these properties. These ``F-singularities of pairs'' correspond tosingularities of pairs of arbitrary codimension in birational geometry. Viathis correspondence, we prove Inversion of Adjunction of arbitrary codimension,which states that for a pair $(X,Y)$ of a smooth variety $X$ and a closedsubscheme $Y \subsetneq X$, if the restriction $(Z, Y|_Z)$ to a normal$\Q$-Gorenstein closed subvariety $Z \subsetneq X$ is klt (resp. lc), then thepair $(X,Y+Z)$ is plt (resp. lc) near $Z$.Comment: 21 pages, AMS-LaTeX; v.2: minor changes, to appear in Invent. Mat

F-singularities of pairs and Inversion of Adjunction of arbitrary codimension