At present, a nonspecialist might well regard the Wiener-Kolmogorov theory of filtering and prediction [1, 2] as "classical' - in short, a field where the techniques are well established and only minor improvements and generalizations can be expected. That this is not really so can be seen convincingly from recent results of Shinbrot [3], Stceg [4], Pugachev [5, 6], and Parzen [7]. Using a variety of methods, these investigators have solved some long-stauding problems in nonstationary filtering and prediction theory. We present here a unified account of our own independent researches during the past two years (which overlap with much of the work [3-71 just mentioned), as well as numerous new results. We, too, use time-domain methods, and obtain major improvements and generalizations of the conventional Wiener theory. In particular, our methods apply without modification to multivariate problems.