We present the first complete next-to-next-to-leading order (NNLO) QCD predictions for differential distributions in the top-quark pair production process at the LHC. Our results are derived from a fully differential partonic Monte Carlo calculation with stable top quarks which involves no approximations beyond the fixed-order truncation of the perturbation series. The NNLO corrections improve the agreement between existing LHC measurements [V. Khachatryan et al. (CMS Collaboration), Eur. Phys. J. C 75, 542 (2015)] and standard model predictions for the top-quark transverse momentum distribution, thus helping alleviate one long-standing discrepancy. The shape of the top-quark pair invariant mass distribution turns out to be stable with respect to radiative corrections beyond NLO which increases the value of this observable as a place to search for physics beyond the standard model. The results presented here provide essential input for parton distribution function fits, implementation of higher-order effects in Monte Carlo generators, as well as top-quark mass and strong coupling determination.