  A partially hyperbolic diffeomorphism $f$ has quasi-shadowing property if for
any pseudo orbit ${x_k}_{k\in \mathbb{Z}}$, there is a sequence of points
${y_k}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ is obtained from
$f(y_k)$ by a motion $\tau$ along the center direction. We show that any
partially hyperbolic diffeomorphism has quasi-shadowing property, and if $f$
has $C^1$ center foliation then we can require $\tau$ to move the points along
the center foliation. As applications, we show that any partially hyperbolic
diffeomorphism is topologically quasi-stable under $C^0$-perturbation. When $f$
has uniformly compact $C^1$ center foliation, we also give partially hyperbolic
diffeomorphism versions of some theorems holden for uniformly hyperbolic
systems, such as Anosov closing lemma, cloud lemma and spectral decomposition
theorem.
