  We prove that non-trivial homoclinic classes of $C^r$-generic flows are
topologically mixing. This implies that given $\Lambda$ a non-trivial
$C^1$-robustly transitive set of a vector field $X$, there is a
$C^1$-perturbation $Y$ of $X$ such that the continuation $\Lambda_Y$ of
$\Lambda$ is a topologically mixing set for $Y$. In particular, robustly
transitive flows become topologically mixing after $C^1$-perturbations. These
results generalize a theorem by Bowen on the basic sets of generic Axiom A
flows. We also show that the set of flows whose non-trivial homoclinic classes
are topologically mixing is \emph{not} open and dense, in general.
