For the former case (Euclidean space), 2-dimensional conformal geometry is simply the Riemann sphere. For three or more dimensions, this is simply inversive geometry (with or without reflections (i.e. inversions!)).

For the latter (Minkowski space) case, in two dimensions, it is simply (taking the universal cover of the compactification), if the space is assumed to be oriented (see Virasoro algebra). This is the default assumption in conformal field theory, the primary field which studies Minkowskilike conformal geometries. For three or more dimensions, its automorphism group is just the SO(n,2).

See also Erlanger program.