  Let $X=(X,\mathcal{B},\mu)$ be a $\sigma$-finite measure space and
\mbox{$f:X\to X$} be a measurable transformation such that the composition
operator $T_f:\varphi\mapsto \varphi\circ f$ is a bounded linear operator
acting on $L^p(X,\mathcal{B},\mu)$, $1\le p<\infty$.
  We provide a necessary and sufficient condition on $f$ for $T_f$ to be
topologically transitive or topologically mixing.
  We also characterize the topological dynamics of composition operators
induced by weighted shifts, non-singular odometers and inner functions.
  The results provided in this article hold for composition operators acting on
more general Banach spaces of functions.
