Coherent states have been introduced in a physical context, first as quasi-classical states in quantum mechanics, then as the backbone of quantum optics and they are described in that spirit in the article Coherent states (see also). However, they have generated a huge variety of generalizations, which have led to a tremendous amount of literature in mathematical physics. In this article, we sketch the main directions of research on this line. For further details, we refer to several existing surveys. A general definition Let be a complex, separable Hilbert space, a locally compact space and a measure on . For each in , denote a vector in . Assume that this set of vectors possesses the following properties: The mapping is weakly continuous, i.e., for each vector in , the function is continuous (in the topology of ). The resolution of the identity holds in the weak sense on the Hilbert space , i.e., for any two vectors in , the following equality holds: A set of vectors satisfying the two properties above is called a family of generalized coherent states. In order to recover the previous definition (given in the article Coherent state) of canonical or standard coherent states (CCS), it suffices to take , the complex plane and Sometimes the resolution of the identity condition is replaced by a weaker condition, with the vectors simply forming a total set in and the functions , as runs through , forming a reproducing kernel Hilbert space. The objective in both cases is to ensure that an arbitrary vector be expressible as a linear (integral) combination of these vectors. Indeed, the resolution of the identity immediately implies that where . These vectors are square integrable, continuous functions on and satisfy the reproducing property where is the reproducing kernel, which satisfies the following properties Some examples We present in this section some of the more commonly used types of coherent states, as illustrations of the general structure given above. Nonlinear coherent states A large class of generalizations of the CCS is obtained by a simple modification of their analytic structure. Let be an infinite sequence of positive numbers (). Define and by convention set . In the same Fock space in which the CCS were described, we now define the related deformed or nonlinear coherent states by the expansion The normalization factor is chosen so that . These generalized coherent states are