The Bondi-Metzner-Sachs group B is the common asymptotic group of all asymptotically flat (lorentzian) space-times, and is the best candidate for the universal symmetry group of general relativity. However, in quantum gravity, complexified or euclidean versions of general relativity are frequently considered, and the question arises: Are there similar symmetry groups for these versions of the theory? In this paper it is shown that there are such analogues of B, and a variety of further ones, either real in any signature, or complex. The relationships between these various groups are described. Irreducible unitary representations (IRS) of the complexification [Note: See the image of page 271 for this formatted text] CB of B itself are analysed. It is proved that all induced IRS of [Note: See the image of page 271 for this formatted text] CB arise from IRS of compact `little groups'. It follows that some IRS of [Note: See the image of page 271 for this formatted text] CB are controlled by the IRS of the `A, D, E' series of finite symmetry groups of regular polygons and polyhedra in ordinary euclidean 3-space. Possible applications to quantum gravity are indicated.