A general necessary condition for the existence of finite dimensional realizations (FDR) of a stochastic bilinear partial differential equation is obtained, providing an easy criterion of non existence of FDR. Comparisons with other finite dimensional filtering concepts are provided. Explicit aind minimal (formal) finite dimensional realizations are then obtained for the class of stochastic linear-quadratic P.D.E.'s (linear time varying operators with coefficients at most quadratic in the variable statje), whose estimation Lie algebra, though infinite dimensional, is finite dimensional at every point The links between FDR's and universal finite dimensional filters (FDF) are displayed by means of gauge transformations and immersions. We show that all the known examples of FDF belong, modulo the above transformations, to the class of stochastic linear-quadratic P.D.E.'s, except the ones corresponding to conditionally gaussian processes [15]