In this chapter, the fault detection issues for nonlinear systems are addressed in the systematic manner. To this end, the coprime factorizations for nonlinear systems, as well as the stable kernel and image representations, as the main control-theoretic tools, are introduced. On this basis, the parameterization form for nonlinear residual generators is studied with the aid of the stable kernel representations. It is followed by the parameterization of nonlinear fault detection systems as well as the threshold setting. However, it is in general difficult to distinguish faults from the nonlinear systems with uncertainties with the aid of the stable kernel representation-based fault detection approaches. Notice that based on the stable image representations, the nominal systems can be modeled as a lower dimensional manifold embedded in the process data space. Then, a projection-based approach is investigated to project the process data onto the image manifold, which promises a higher dimensional classification (both in the input and output subspaces) than the stable kernel representation-based fault detection (in the output subspace). It enables a full decoupling of the nominal and abnormal system dynamics in the input and output data space, and can be implemented for optimal fault detection in nonlinear systems with uncertainties.