  In this work we study the class of mostly expanding partially hyperbolic
diffeomorphisms. We prove that such class is $C^r$-open, $r>1$, among the
partially hyperbolic diffeomorphisms (in the narrow sense) and we prove that
the mostly expanding condition guarantee the existence of physical measures and
provide more information about the statistics of the system. Ma\~n\'e's
classical derived-from-Anosov diffeomorphism on $\mathbb{T}^3$ belongs to this
set.
