The paper characterizes all linear function observers for descriptor systems. With the function observers formulated in the frequency domain, the problem of finding all function observers is reduced to that of solving a proper stable linear matrix equation. Parametrization of its general solution results in the desired characterization of the function observers. These function observers are then realized in the state space and it is shown that if an appropriate state space transformation is introduced, the class of the reduced order observers can also be derived through this realization.