  We prove that every globally $F$-regular variety is log Fano. In other words,
if a prime characteristic variety $X$ is globally $F$-regular, then it admits
an effective $\bQ$-divisor $\Delta$ such that $-K_X - \Delta$ is ample and $(X,
\Delta)$ has controlled (Kawamata log terminal, in fact globally $F$-regular)
singularities. A weak form of this result can be viewed as a prime
characteristic analog of de Fernex and Hacon's new point of view on Kawamata
log terminal singularities in the non-$\bQ$-Gorenstein case. We also prove a
converse statement in characteristic zero: every log Fano variety has globally
$F$-regular type. Our techniques apply also to $F$-split varieties, which we
show to satisfy a "log Calabi-Yau" condition. We also prove a Kawamata-Viehweg
vanishing theorem for globally $F$-regular pairs.
