An affine representation of a topological (Lie) group G is a continuous (smooth) homomorphism from G to the automorphism group of an affine space, A.
An example is the action of the Euclidean group E(n) upon the Euclidean space En.
Since the affine group in dimension n is a matrix group in dimension n + 1, an affine representation may be thought of as a particular kind of linear representation. We may ask whether a given affine representation has a fixed point in the given affine space A. If it does, we may take that as origin and regard A as a vector space: in that case, we actually have a linear representation in dimension n. This reduction depends on a group cohomology question, in general.
See also projective representation, group action.