||Piecewise-smooth (PWS) systems are used to model many important physical processes, such as impacting, friction, switching, and sliding systems. In this talk bifurcation theory and some applications for PWS systems are investigated. The problem of homoclinic bifurcations in planar continuous piecewise-linear (PWL) systems with two zones is particularly studied. The systems with homoclinic orbits can be divided into two cases: the visible saddle-focus (or saddle-center) case and the case of twofold nodes with opposite stability. Two kinds of homoclinic bifurcations are discussed: one is generically related to nondegenerate homoclinic orbits; the other is the discontinuity induced homoclinic bifurcations related to the boundary. The results show that at least two parameters are needed to unfold all possible homoclinc bifurcations in the systems. Other research works related to PWS systems in both theory and applications are mentioned.