This paper is concerned with the analysis and classification of non-standard bifurcations in piecewise smooth feedback systems. Bifurcations involving fixed points in maps, equilibria and limit cycles in flows are considered with particular attention to: border collisions of fixed points in maps; non-smooth bifurcations of equilibria, grazing bifurcations and sliding bifurcations of limit cycles in flows. The aim is to describe existing and novel results to form the basis of a consistent theory of bifurcations in such systems. In so doing, a novel approach to classify non-smooth bifurcations of equilibria in flows is presented. Experimental results on pulse-width modulated voltage-mode controlled DC/DC power converters motivates the relevance to applications of the analysis proposed.