Function spaces play an important role in complex analysis, in the theory of differential equations, in functional analysis and in almost every other branch of modern mathematics. Let FC(Y,Z) be the set of all fuzzy continuous functions from a fuzzy topological space Y into a fuzzy topological space Z. Our aim in this paper is to study the notion of group, fuzzy group, topological group, and fuzzy topological group on the (fuzzy) function space FC(Y,Z).