Let (X, .) be a semigroup, (Y, +) an abelian group and g:X→Y. The first and second order Cauchy differences of g are $D^1 g (a,b) = g (ab) - g (a) - g (b)$, $D^2 g (a,b,c) = g (abc) - g (ab) - g(bc) - g (ac)+ g(a)+ g (b)+ g (c)$. Finite order Cauchy differences $D^n f$ are defined recursively. In the case of Y = Z, a ring where multiplication is distributive over addition, we show that functions g : X→ Z with finite Cauchy differences are closed under multiplication. The equation $D^n g = 0$ is considered for finite abelian groups.