A number of different fundamental problems in fault detection and fault identification are formulated in this paper. The fundamental problems include exact, almost, generic and class-wise fault detection and identification. Necessary and sufficient conditions for the solvability of the fundamental problems are derived. These conditions are weaker than the ones found in the literature since we do not assume any particular structure for the residual generator. At the end of the paper, a time domain synthesis procedure based on state-space methods to construct appropriate residual generators is given. Copyright (C) 2000 John Wiley & Sons, Ltd.