  We strengthen the Carleson-Hunt theorem by proving $L^p$ estimates for the
$r$-variation of the partial sum operators for Fourier series and integrals,
for $p>\max\{r',2\}$. Four appendices are concerned with transference, a
variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear
Fourier transforms and ergodic theory.
