Introduction. There have been several papers written on the subject of determining all homomorphisms, and more particularly, isomorphisms of group algebras of locally compact abelian groups. Except in certain special cases there does not seem to be much known. One case which has been treated is the case of two groups G and H, where H has a connected dual group L . In this case it has been shown that the only homomorphisms of L1(G) into L1(H) are essentially those induced by homomorphisms of G into H [1]. This result was proved in the case where H is the real line and by means of the structure theory of locally compact abelian groups, extended to the more general case. The crucial point in the proof seems to be the obvious fact that a Fourier-Stieltjes transform taking only the values zero and one must either be identically zero or identically one. Equivalently we may say that the only idempotent measures on H are the zero measure, and Haar measure of the identity subgroup. Another case, that in which H is the circle group, has been solved in [5], [6]. In this case too, the complete analysis of -idempotent measures on the circle achieved in [4], was very heavily used. The author in a previous paper [2], has determined all the idempotent measures on locally compact groups. Thus, it seems reasonable that one should now be able to completely solve the homomorphism problem. In this paper, we shall do exactly that. A very simple, but hitherto unnoticed, relationship between the homomorphism problem and idempotent measures is established in the case of compact G and H. Then a passage to the Bohr compactifications of the groups in question yields the general result. There are certain technical complications which appear, some of which are standard, such as convolutions with approximate identities, which we hope will not confuse the reader. It is perhaps unnecessary to add that at all times the reader should bear in mind the concrete examples of Fourier series and Fourier integrals to better understand what is happening. In the case of Fourier series, our problem is precisely one of determining which mappings of Fourier coefficients in m-variables into coefficients in n-variables, send Fourier coefficients into Fourier coefficients. More specifically, let 7r be a