Abstract. A smooth affine surface X defined over the complex field C is an ML 0 surface if the MakarLimanov invariant ML(X) is trivial. In this paper we study the topology and geometry of ML 0 surfaces. Of particular interest is the question: Is every curve C in X which is isomorphic to the affine line a fiber component of an A 1 -fibration on X? We shall show that the answer is affirmative if the Picard number ρ(X) = 0, but negative in case ρ(X) ≥ 1. We shall also study the ascent and descent of the ML 0 property under proper maps.

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