This is a textbook intended for a one-quarter (or one-semester, depending on the pace) course at the graduate level in Engineering. The prerequisites are Elementary State-Space theory and Elementary (second-order Gaussian) Stochastic Process theory. As a textbook, it does not not purport to be a compendium of all known work on the subject. Neither is it a "trade book." Rather it attempts a logically sequenced set of topics of proven pedagogical value, emphasizing theory while not devoid of practical utility. The organization is based on experience gained over a period of ten years of class room teaching. It develops those aspects of Kalman Filtering lore which can be given a firm mathematical basis, avoiding the industry syndrome manifest in professional short courses: "Here is the recipe. Use it, it will "work"!"