  It is well known that smooth (or continuous) vector fields cannot be
topologically transitive on the sphere $\S^2$. Piecewise-smooth vector fields,
on the other hand, may present non-trivial recurrence even on $\S^2$.
Accordingly, in this paper the existence of topologically transitive
piecewise-smooth vector fields on $\S^2$ is proved, see Theorem
\ref{teorema-principal}. We also prove that transitivity occurs alongside the
presence of some particular portions of the phase portrait known as {\it
sliding region} and {\it escaping region}. More precisely, Theorem
\ref{main:transitivity} states that, under the presence of transitivity,
trajectories must interchange between sliding and escaping regions through
tangency points. In addition, we prove that every transitive piecewise-smooth
vector field is neither robustly transitive nor structural stable on $\S^2$,
see Theorem \ref{main:no-transitive}. We finish the paper proving Theorem
\ref{main:general} addressing non-robustness on general compact two-dimensional
manifolds.
