The state-space realization problem is a very basic and fundamental problem of control theory. The topic is also becoming increasingly important as practitioners of both physical and social sciences find it crucial to model very complex systems based on input-output data only. In this thesis, a review of the topic will be given for general nonlinear systems and for the less general linear case as well. The thesis will also present some new theoretical results that contribute to the development of the state-space realization topic. Specifically, an important result will show that if a system can be identified by an input-output equation of a particular form, which is fairly general, then a state-space realization can always be easily derived directly from the input-output map. Finally, the theory will be applied to find a state-space model for a nonlinear hydraulic system based on its input-output data.