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It is by now well known that the standard local (i.e., pointwise) energy conditions always can be violated in quantum field theory in curved (and flat) spacetime, even when these energy conditions hold for the corresponding classical field. Nevertheless, some global constraints on the stress-energy tensor may exist. Indeed recent work has shown that the averaged null energy condition (ANEC), which requires the positivity of energy suitably averaged along null geodesics, holds for a wide class of states of a minimally coupled scalar field on Minkowski spacetime, and also (in the massless case) on a wide class of states in curved two-dimensional spacetimes satisfying certain asymptotic regularity properties. In this paper, we strengthen these results by proving that, for the massless scalar field in an arbitrary globally hyperbolic two-dimensional spacetime, the ANEC holds for all Hadamard states along any complete, achronal null geodesic. In our analysis, the general, algebraic notion of "state" is used, so, in particular, it is not even assumed that our state belongs to any Fock representation. Our proof shows that the ANEC is a direct consequence of the general positivity condition which must hold for the two-point function of any state. Our results also can be extended (with a restriction on states) to the massive scalar field in twodimensional Minkowski spacetime and (with an additional restriction on states) to the (massless or massive) minimally coupled scalar field on four-dimensional Minkowski spacetime. In the case of a (curved) four-dimensional spacetime with a bifurcate Killing horizon, our proof also extends to establish the ANEC for the null geodesic generators of the horizon (provided that there exists a stationary Hadamard state of the field). This latter result implies that the ANEC must hold for the massive Klein-Gordon field in de Sitter spacetime.