Polynomials on groups play an important role in the theory of functional equations because in most cases the general solution of a linear functional equation can be described by polynomials. For example, the only continuous solution of the square functional equation \(f(x+t,y+t)+f(x+t,y-t)+f(x-t,y-t)+f(x-t,y+t)=4f(x,y),\) where \(f:R\times R\to R\) and x, y, t are real variables, is a polynomial in x and y [see \textit{J. Aczél,} the reviewer, \textit{M. A. McKiernan} and \textit{G. N. Saković,} Aequationes Math. 1, 37-53 (1968; Zbl 0157.461)]. In this paper the author extends the main results of the classical theory to a local direction by introducing the notion of local polynomials on topological Abelian groups. By using this notion he solves a general local functional equation and applies his results to a local mean value property.