Let X and Y be Banach spaces and S a subset of the space of (linear, continuous) operators from X to Y . We say that an operator T belongs locally to S if for every x ∈ X there is S ∈ S, possibly depending on x, such that Tx = Sx. ‘Pointwise’ should be better than ‘locally’, but we have followed tradition. If each operator that belongs locally to S belongs in fact to S we say that S is algebraically reflexive. When Y = X and S = Iso(X) is the group of isometries of X we say that T is a local isometry of X. (In this paper ‘isometry’ means linear surjective isometry.) Similarly, a local automorphism of a Banach algebra is an operator which agrees at every point with some automorphism. Also, we will consider approximate local isometries and automorphisms. These are operators having with the following property: given x ∈ X and e > 0, there is an isometry (respectively, an automorphism) S of X such that ‖Tx− Sx‖ < e. The study of local isometries and automorphisms of Banach algebras spurred a considerable interest in recent years (see the bibliography of the dissertation [16]). In this paper we deal with local isometries and automorphisms of the algebras C0(L). As usual, we write C0(L) or C K 0 (L) for the Banach algebra of all continuous K-valued functions on the locally compact space L vanishing at infinity, where K is either C or R. If L is compact the subscript will be omitted. By the Banach-Stone theorem, if T is local isometry of C0(L), then for each f there are a homeomorphism φ of L and a continuous unitary u : L→ K such that Tf = u(f ◦ φ).