We describe both the key principles and real application examples of a unified theory which allows us to perform the on-board incipient fault detection and isolation tasks involved in monitoring for condition-based maintenance. This theory is known under the name of local approach, and it is especially suited to component faults. It aims at transforming complex detection problems concerning a parameterized stochastic process into the universal problem of monitoring the mean of a Gaussian vector. Based on a small deviation assumption, the key tools are the first-order Taylor expansion and the asymptotic Gaussianity of a convenient parameter estimating function. ml,ls, iv and subspace identification methods are addressed in this perspective. In the case of dynamic processes given in state-space form,the approach may also call for observer-based slate estimation or state elimination. Experiments concerning both simulated and real data, for linear and nonlinear dynamical processes, are reported. How the key principles and features of the local approach compare with those of other approaches is discussed.