Observers cire used to estimate the states of dynamiccil systems whenever are not available through direct measurements. Although the design of linear observers is a well-developed branch of control theory, its counterpart for nonlinear systems is a relatively new field. In this thesis, an observer construction methodology is proposed lor a chiss of nonlinear systems satisfying certain conditions. Then, the problem of synchronizing chaotic systems, which has found recent appliccitions in the area of secure message transmission, is addressed from the observer design point of view. In the design, we exploited one of the essential properties of the chaotic systems that the trajectories remain in a bounded region of the state space. It is also shown that, for certciin well-known chaotic systems, the system structure enables one to nse linecir observer schemes in order to have global synchronization.