  Pugh and Shub have conjectured that essential accessibility implies
ergodicity, for a $C^2$, partially hyperbolic, volume-preserving
diffeomorphism. We prove this conjecture under a mild center bunching
assumption, which is satsified by all partially hyperbolic systems with
1-dimensional center bundle. We also obtain ergodicity results for
$C^{1+\gamma}$ partially hyperbolic systems.
