It is known that the identity component of the automorphism group of a projective algebraic variety is an algebraic group. This is not true in general for quasi-projective varieties. In this note we address the question: given an affine algebraic surface Y , as to when the identity component Aut • (Y) of the automorphism group Aut(Y) is an algebraic group? We show that this happens if and only if Y admits no effective action of the additive group. In the latter case, Aut • (Y) is an algebraic torus of rank ≤ 2.