Given two groups \(G,H\), the functional equation \[ f(xy)+f(xy^{-1})=2f(x)+2f(y),\qquad x,y\in G \tag{(1)} \] is called the quadratic functional equation, where \(f:G\to H\) is considered as an unknown function. Assuming that \(G\) and \(H\) are abelian and \(H\) is uniquely 2-divisible, \textit{J. Aczél} [Period. Math.-Phys. Astron., II. Ser. 20, 65--73 (1965; Zbl 0151.20904)] proved that the solutions of (1) are quadratic functions, i.e., they are of the form \(f(x)=Q(x,x)\), where \(Q:G\times G\to H\) is a symmetric bimorphism. The authors of this paper investigate the quadratic equation on groups \(G\) that are not necessarily abelian. One of the main results of the paper shows that any solution of (1) is a function on the factor group \(G/[G,[G,G]]\). (Here \([A,B]\) denotes the subgroup generated by the elements \([a,b]=aba^{-1}b^{-1}\) \(a\in A,\,b\in B\)). Another main result states that the solutions of (1) are quadratic functions provided \([G,[G,G]]=[G,G]\). Finally the cases when \(G\) is the Heisenberg group or the general linear group \(GL(n,R)\) are discussed in details.