We study observer design and output feedback stabilization of switched and nonlinear time varying systems. For stabilization of delayed switched systems via an observer, we develop a new extension of a recently proposed trajectory based approach which is fundamentally different from classical Lyapunov function based methods. This new extension of trajectory based approach can be applied to a wide range of systems with time varying delays and it tackles the issue of finding appropriate Lypunov functions to establish the stability of delayed switched systems. Our stabilization methodology does not require stabilizability and detectability of all of the subsystems of the switched system and we do not impose any constraint on the derivative of the time varying delay. For nonlinear time varying systems, we build a new type of finite-time smooth observer in the case where a state dependent disturbance affects the linear approximation. We combine this finite time observer design and a switched systems approach to develop stabilizing feedbacks for nonlinear time varying systems whose outputs are only available on some finite time intervals. Again, we use an extension of the trajectory based approach to establish the stability of the closed-loop system. Motivated by the fact that the measured components of the state do not need to be estimated, we also construct reduced order finite time observers for a broad class of nonlinear time-varying systems. We show how these reduced order finite time observers can be used to solve dynamic output feedback stabilization problem for MIMO nonlinear time varying systems. Finally, we design a finite time observer to estimate the exact state of a continuous-time linear time varying system from sampled output in the presence of a piecewise continuous disturbance.