Observers are needed to estimate unmeasured state variables of dynamic systems. They are often used for output feedback control design when only the outputs are available for the control design. Observers can also be used for other estimation purposes such as fault detection and diagnostics. There are many results on nonlinear observer design in literature, and in this chapter, we can introduce only a number of results. Observer design for linear systems is briefly reviewed before the introduction of observers with linear error dynamics. We then introduce another observer design method based on Lyapunov's auxiliary theorem, before the observer design for systems with Lipschitz nonlinearities. At the end of this chapter, adaptive observers are briefly described.