The dynamic properties of dense two-dimensional (2D) polymer melts are studied using discontinuous molecular dynamics simulations. Both strictly 2D and quasi-2D systems are investigated. The strictly 2D model system consists of a fluid of freely jointed tangent hard disc chains. The translational diffusion coefficient, D, is strongly system size dependent with D ∼ ln L where L is the linear dimension of the square simulation cell. The rotational correlation time, τrot, is, however, independent of system size. The dynamics is consistent with Rouse behavior with D/ln L ∼ N−1 and τrot ∼ N2 for all area fractions. Analysis of the intermediate scattering function, Fs(k, t), shows that the dynamics becomes slow for N = 256 and the area fraction of 0.454 and that there might be a glass transition for long polymers at sufficiently high area fractions. The polymer mobility is not correlated with the conformation of the molecules. In the quasi-2D system hard sphere chains are confined between corrugated surfaces so that chains cannot go over each other or into the surfaces. The conformational properties are identical to the 2D case, but D and τrot are independent of system size. The scaling of D and τrot with N is similar to that of strictly 2D systems. The simulations suggest that 2D polymers are never entangled and follow Rouse dynamics at all densities.