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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.