Let X=(X,B,μ) be a σ-finite measure space and f:X→X be a measurable transformation such that the composition operator Tf:φ↦φ∘f is a bounded linear operator acting on Lp(X,B,μ), 1≤p<∞. We provide a necessary and sufficient condition on f for Tf 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.