Top quark-antiquark (t tbar) pairs will be produced copiously at the Tevatron collider and in huge numbers at the LHC. This will make possible detailed investigations of the properties and interactions of this quark flavor. The analysis and interpretation of future data requires precise predictions of the hadronic production of ttbar pairs and of their subsequent decays. In this talk the reactions p pbar, p p --> t tbar X --> l^+l^-X are considered and results are presented of our calculation of the dilepton angular distribution at next-to-leading order QCD, keeping the full dependence on the spins of the intermediate t tbar state. The angular distribution is determined for different choices of reference axes that can be identified with the t and tbar spin axes. While the QCD corrections to the leading-order distribution turn out to be small in the case of the LHC, we find them to be sizeable in the case of the Tevatron and find, moreover, the angular distribution to be sensitive to the parton content of the proton.