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The Euler characteristic of a 2-dimensional compact manifold and the localbehavior of smooth vector fields defined on it are related to each other bymeans of the Poincar\'e--Hopf Theorem. Despite of the importance of Filippovvector fields, concerning both their theoretical and applied aspects, untilnow, it was not known if this result it still true for Filippov vector fields.In this paper, we show it is.  While in the smooth case the singularities consist of the points where thevector field vanishes, in the context of Filippov vector fields the notion ofsingularity also comprehend new kinds of points, namely, pseudo-equilibria andtangency points. Here, the classical index definition for singularities ofsmooth vector fields is extended to singularities of Filippov vector fields.Such an extension is based on an invariance property under a regularizationprocess. With this new index definition, we provide a version of thePoincar\'e--Hopf Theorem for Filippov vector fields. Consequently, we also geta Hairy Ball Theorem in this context, i.e. ``any Filippov vector field definedon a sphere must have at least one singularity (in the Filippov sense)''.

Poincar\'e-Hopf Theorem for Filippov vector fields on 2-dimensional compact manifolds