An overview of new results on the sensitivity analysis of two categories of hybrid system is presented. Firstly, a theory is presented for local, first-order sensitivity analysis of limit-cycle oscillating hybrid systems, which are dynamical systems exhibiting both continuous-state and discrete-state dynamics whose state trajectories are closed, isolated and time-periodic. A method is described for decomposition of the parametric sensitivities into three parts, characterizing the influence of parameter changes on period, state variable amplitudes, and relative phases, respectively. The computation of parametric sensitivities of period, amplitudes, and phases is also described. It is shown that, in general, it is incorrect to set the initial conditions for parametric sensitivities of state variables to zero. Secondly, using results from nonsmooth analysis, sufficient conditions are presented for the existence of the forward and adjoint sensitivities of parametric ordinary differential equations with locally Lipschitz continuous vector fields. Unique aspects of these results are demonstrated via examples.