In history, physics, and ordinary life, we often explain a pattern of correlations between distant events in terms of some linking event in their common past. This kind of explanation is strongly time-asymmetric, for we take for granted that there are no irreducibly "teleological" correlations, needing to be explained in terms of some linking event in their common future. What sort of fact about the world is it that it obeys this time-asymmetric "No Teleology Principle"? A natural hypothessis is that it is tied to the time-asymmetry of thermodynamics--i.e., that the thermodynamic asymmetry provides the objective feature of our world, in virtue of which the No Teleology Principle is true. However, it turns out that some intuitively plausible applications of the No Teleology Principle in microphysics are not compatible with this hypothesis. This suggests that either (i) the hypothesis is mistaken, and there is a further objective time-asymmetry in the physical world, in addition to that of thermodynamics; or (ii) these applications of the No Teleology Principle in microphysics are misguided. I'll argue that (i) runs counter to the apparent T-symmetry of the laws of physics, and that (ii) seems to offer considerable benefits in quantum mechanics. We thus have two reasons for taking seriously the possibility that the No Teleology Principle is an unreliable intuition in microphysics. The paper covers similar ground to Chapter 5 of my book Time's Arrow and Archimedes' Point (OUP, 1996), but the arguments have improved, and I respond to some objections.