We start with the Heisenberg–Weyl algebra and after the definitions of the Fock states we give the definition of the coherent state of this group. This is followed by the exposition of the SU(2) and SU(1, 1) algebras and their coherent states. From there we go on describing the binomial state and its extensions as realizations of the SU(2) group. This is followed by considering the negative binomial states, and some squeezed states as realizations of the SU(1, 1) group. Generation schemes based on physical systems are mentioned for some of these states.