Summary: We present behavioral existence and parametrization results for input observers of IO (input/output) behaviors and for pseudo state observers of Rosenbrock equations, i.e., of systems given by polynomial matrix descriptions. Our results significantly extend those of Wolovich from 1974. Valcher and Willems started the behavioral theory of observers in 1999 and Fuhrmann treated all aspects of observers in a recent comprehensive paper. We use the (behavioral) observers and associated error behaviors of these authors, but in contrast to them require the observers to be IO behaviors which are proper, but not necessarily consistent. Our results are also applicable to their more general behaviors and, conversely, their theorems are applicable to our situations. More recently Bisiacco, Valcher and Willems also considered non-consistent dead-beat observers.We discuss the relation of our work to that of our predecessors in some detail. The \(T\) in the title refers to a multiplicatively closed set of ordinary differential or shift operators in the standard cases, gives rise to \(T\)-autonomy, \(T\)-stability and \(T\)-observers and enables the simultaneous study of tracking, asymptotic, dead-beat, exact and other observers both in the continuous and the discrete cases. We derive new algorithms for the construction of proper \(T\)-observers and apply them in an instructive example, computed with MAPLE. Our proofs rely on module-behavior duality and on linear algebra over the ring of proper and \(T\)-stable rational functions.