F-singularities via alterations

For a normal F-finite variety X and a boundary divisor \Delta we give a uniform description of an ideal which in characteristic zero yields the multiplier ideal, and in positive characteristic the test ideal of the pair (X,\Delta). Our description is in terms of regular alterations over X, and one consequence of it is a common characterization of rational singularities (in characteristic zero) and F-rational singularities (in characteristic p) by the surjectivity of the trace map \pi_* \omega_Y \to \omega_X for every such alteration \pi \: Y \to X.
Furthermore, building on work of Bhatt, we establish up-to-finite-map versions of Grauert-Riemenscheneider and Nadel/Kawamata-Viehweg vanishing theorems in the characteristic p setting without assuming W2 lifting, and show that these are strong enough in some applications to extend sections.