This paper analyzes the observer design problem in the behavioral context. Observability and detectability notions are first introduced and fully characterized. Necessary and sufficient conditions for the existence of an observer, possibly an asymptotic or an exact one, are introduced, and a complete parameterization of all admissible observers is given. The problem of obtaining observers endowed with a (strictly) proper transfer matrix and the design of observer-based controllers are later addressed and solved. Finally, the above issues are particularized to the case of state-space systems, thus showing they naturally generalize well-known theorems of traditional system theory.