The well-known state-space paradigm for the analysis of systems with switching dynamics has been predominant in control theory. However, in many cases this framework is not justified from the physical point of view, since first principles models do not necessarily satisfy a global first-order differential structure. An example of this situation can be easily found when the changing laws of a switched system do not share the same state space, e.g. a power converter with multiple (dis-)connectable loads. Moreover, in the analysis of complex electrical networks, first principle models are more satisfactorily represented by impedances which naturally lead to sets of higher-order differential equations. Prompted by these issues, we have developed a new approach to switched systems based on behavioral system theory which is a trajectory-oriented rather than a represention-based approach. This framework permits to accomodate first principle equations, possibly of higher-order, without resorting to the use of pre-defined mathematical structures and thus circumventing the need to add fictitious variables and equations to analyze the system. Modeling, stability analysis, control and applications in power electronics are currently the main objectives in our scientific agenda.