  In this paper the asymptotic behavior of trajectories of discontinuous vector
fields is studied. The vector fields are defined on a two-dimensional
Riemannian manifold $M$ and the confinement of trajectories on some suitable
compact set $K$ of $M$ is assumed. The behavior of the global trajectories is
fully analyzed and their limit sets are classified. The presence of limit sets
having non-empty interior is observed. Moreover, the existence of the so called
sliding motion is allowed on $M$. The results contemplate a list of possible
limit sets as well the existence of non-recurrent dynamics and the presence of
nondeterministic chaos. Some examples and classes of systems fitting the
hypotheses of the main theorems are also provided in the paper.
