We describe the nonzero temperature (T ), low frequency (ω) dynamics of the order parameter near quantum critical points in two spatial dimensions (d), with a special focus on the regimehω ≪ k B T . For the case of a 'relativistic', O(n)-symmetric, bosonic quantum field theory we show that, for small ǫ = 3 − d, the dynamics is described by an effective classical model of waves with a quartic interaction. We provide analytical and numerical analyses of the classical wave model directly in d = 2. We describe the crossover from the finite frequency, "amplitude fluctuation", gapped quasiparticle mode in the quantum paramagnet (or Mott insulator), to the zero frequency "phase" (n ≥ 2) or "domain wall" (n = 1) relaxation mode near the ordered state. For static properties, we show how a surprising, duality-like transformation allows an exact treatment of the strong-coupling limit for all n. For n = 2, we compute the universal T dependence of the superfluid density below the Kosterlitz-Thouless temperature, and discuss implications for the high temperature superconductors. For n = 3, our computations of the dynamic structure factor relate to neutron scattering experiments on La 1.85 Sr 0.15 CuO 4 , and to light scattering experiments on double layer quantum Hall systems. We expect that closely related effective classical wave models will apply also to other quantum critical points in d = 2. Although computations in appendices do rely upon technical results on the ǫ-expansion of quantum critical points obtained in earlier papers, the physical discussion in the body of the paper is self-contained, and can be read without consulting these earlier works.