Given a finite abelian group V and subsets S1, S2, ... ,Sn of V, write Ti for the totality of all the possible differences between elements of Si (with repetitions counted multiply) and T for the totality of members of all the Ti. If T contains each non-zero element of V the same number of times, then the sets S1, S2,...,Sn will be called supplementary difference sets. We discuss some properties for such sets, give some existence theorems and observe their use in the construction of Hadamard matrices and balanced incomplete block designs.