The filtering and smoothing problems are studied for a class of distributed-parameter information and control systems described by linear partial differential or integro-differential equations. The results are first derived in discrete-time form following a Bayesian information processing approach and are then converted in continuous-time form using an extension of Kalman's limiting procedure. The paper concludes by showing how the results should be formally used for treating nonlinear systems. The filters derived here are the most general and involve the distributed-parameter filters derived by other authors as special cases.