Filtering theory is central to a wide variety of disciplines, including control, communications, and signal processing. This monograph treats linear filtering and inverse filtering in a general setting, presenting a state-of-the art view of H_2, H_8 and a variety of mixed H_2 and H_8 filtering issues. Applications are also treated. Key strengths of the book: - Develops a general theory of H_2 filtering, H_2 delayed filtering, and H_2 inverse filtering or H_2 deconvolution in a broad framework with a complete analysis, as well as synthesis aspects of filter design - Thorough examination of several hierarchically ordered layers of noise decoupled filtering problems, delayed filtering problems, and deconvolution problems under a single umbrella - Illustrates the application of filtering theory to fault detection, isolation, and estimation - Numerous examples with MATLAB, comprehensive bibliography, and index - Open problems The book will serve as an advanced graduate text for a course in filtering in applied mathematics or electrical engineering departments. It will also be an invaluable resource for practicing engineers in industry.