In this article we describe some qualitative and geometric aspects of nonsmooth dynamical systems theory around typical singularities. We also establish an interaction between nonsmooth systems and geometric singular perturbation theory. Such systems are represented by discontinuous vector fields on Rl, l ≥ 2, where their discontinuity set is a codimension one algebraic variety. By means of a regularizaron process proceeded by a blow-up technique we are able to bring about some results that bridge the space between discontinuous systems and singularly perturbed smooth systems. We also present an analysis of a subclass of discontinuous vector fields that present transient behavior in the 2-dimensional case, and we dedicate a section to providing sufficient conditions in order for our systems to have local asymptotic stability. © 2009 Society for Industrial and Applied Mathematics.81508526ALEXANDER, J.C., SEIDMAN, T.I., Sliding modes in intersecting switching surfaces (1998) I. Blending, Houston J. Math, 24, pp. 545-569ALEXANDER, J.C., SEIDMAN, T.I., Sliding modes in intersecting switching surfaces (1999) II. Hysteresis, Houston J. Math, 25, pp. 185-211BROUCKE, M., PUGH, C., SIMIC, S., Structural stability of piecewise smooth systems (2001) Comput. Appl. Math, 20, pp. 51-89BUZZI, C., SILVA, P.R., TEIXEIRA, M.A., A singular approach to discontinuous vector fields on the plane (2006) J. Differential Equations, 231, pp. 633-655DI BERNARDO, M., BUDD, C., CHAMPNEYS, A., KOWALCZYK, P., (2008) Piecewise-Smooth Dynamical Systems. Theory and Applications, , Appl. Math. Sci. 163, Springer-Verlag, LondonDUMORTIER, F., ROUSSARIE, R., Canard cycles and center manifolds (1996) Mem. Amer. Math. Soc, 121, p. 132708FENICHEL, N., Geometric singular perturbation theory for ordinary differential equations (1979) J. Differential Equations, 31, pp. 53-98FILIPPOV, A.F., (1988) Differential Equations with Discontinuous Righthand Sides, , Math. Appl, Soviet Ser, Kluwer Academic Publishers, Dordrecht, The NetherlandsC. JONES, Geometric singular perturbation theory, in Dynamical Systems, C.I.M.E. Lectures (Montecatini Terme, 1994), Lecture Notes in Math. 1609, Springer-Verlag, Heidelberg, 1995, pp. 44-118LLIBRE, J., SILVA, P.R., TEIXEIRA, M.A., Regularization of discontinuous vector fields via singular perturbation (2007) J. Dynam. Differential Equations, 19, pp. 309-331LLIBRE, J., SILVA, P.R., TEIXEIRA, M.A., Sliding vector fields via slow-fast systems (2008) Bull. Belg. Math. Soc. Simon Stevin, 15, pp. 851-869LLIBRE, J., TEIXEIRA, M.A., Global asymptotic stability for a class of discontinuous vector fields in R2 (2007) Dyn. Syst, 22, pp. 133-146TEIXEIRA, M.A., Stability conditions for discontinuous vector fields (1990) J. Differential Equations, 88, pp. 15-29TEIXEIRA, M.A., Perturbation theory for non-smooth dynamical systems Encyclopedia of Complexity and Systems Science, , G. Gaeta, ed, Springer-Verlag, New York, to appear