  A partially hyperbolic diffeomorphism $f$ is structurally quasi-stable if for
any diffeomorphism $g$ $C^1$-close to $f$, there is a homeomorphism $\pi$ of
$M$ such that $\pi\circ g$ and $f\circ\pi$ differ only by a motion $\tau$ along
center directions. $f$ is topologically quasi-stable if for any homeomorphism
$g$ $C^0$-close to $f$, the above holds for a continuous map $\pi$ instead of a
homeomorphism. We show that any partially hyperbolic diffeomorphism $f$ is
topologically quasi-stable, and if $f$ has $C^1$ center foliation $W^c_f$, then
$f$ is structurally quasi-stable. As applications we obtain continuity of
topological entropy for certain partially hyperbolic diffeomorphisms with one
or two dimensional center foliation.
