  We study dynamical systems which have bounded complexity with respect to
three kinds metrics: the Bowen metric $d_n$, the max-mean metric $\hat{d}_n$
and the mean metric $\bar{d}_n$, both in topological dynamics and ergodic
theory.
  It is shown that a topological dynamical system $(X,T)$ has bounded
complexity with respect to $d_n$ (resp. $\hat{d}_n$) if and only if it is
equicontinuous (resp. equicontinuous in the mean). However, we construct
minimal systems which have bounded complexity with respect to $\bar{d}_n$ but
not equicontinuous in the mean.
  It turns out that an invariant measure $\mu$ on $(X,T)$ has bounded
complexity with respect to $d_n$ if and only if $(X,T)$ is
$\mu$-equicontinuous. Meanwhile, it is shown that $\mu$ has bounded complexity
with respect to $\hat{d}_n$ if and only if $\mu$ has bounded complexity with
respect to $\bar{d}_n$ if and only if $(X,T)$ is $\mu$-mean equicontinuous if
and only if it has discrete spectrum.
