Dynamics of a curved flame front in cylindrical tubes is studied by means of numerical simulation of the complete system of hydrodynamic equations including thermal conduction, fuel diffusion, viscosity and chemical kinetics. The complete three-dimensional system of equations is reduced to a two-dimensional system with the account of cylindrical symmetry of the problem. The dependence of the flame velocity on the tube diameter and the expansion coefficient of the fuel is investigated. It is obtained that the velocity increase due to the curved shape of the flame front is considerably larger for the case of cylindrical tubes compared to the two-dimensional curved flames. The regime of strong initiation of the flame instability is obtained for narrow tubes, when all perturbation modes of small amplitude are stable, but a curved stationary flame is still possible. The simulation results indicate that there is no maximal velocity for curved flames in cylindrical tubes unlike two-dimensional flames. Instead the flame velocity increases with no limit with the increase of the tube diameter. The case of a flame propagating upwards in a gravitational field is also investigated.