The classical almost periodic compactification of a locally compact group G can be defined to be the largest compact group which contains a dense homomorphic image of G. This has an interpretation in terms of the Banach algebra L 1 (G), and certain module maps from L 1 (G) to its dual L ∞(G), which are compact. This definition obviously makes sense for any Banach algebra, and, in particular, has been studied for the Fourier algebra A(G). By analogy with the L 1 (G) case, and arguing by duality, we might expect that the almost periodic compactification of A(G) should be a group C∗ -algebra of some discrete group. This can be made precise by applying recent work of Soltan on compacitifications in the quantum group setting. The classical definition does not always yield this, however. We shall argue for some different definitions, making use of operator spaces, which do yield an object which can be regarded as a compactification.