Abstract : We simulate the Boussinesq equations for Rayleigh-Bénard convection in a cylindrical container. In the first part, for aspect ratios near 1.5, Prandtl number 1 and insulating sidewalls, the transition from an axisymmetric stationary flow to time-dependent flows is studied using nonlinear simulations, linear stability analysis and bifurcation theory. At a Rayleigh number near 25000, the axisymmetric flow becomes unstable to standing or travelling azimuthal waves. The standing waves are slightly unstable to travelling waves. This scenario is identified as a Hopf bifurcation in a system with $O(2)$ symmetry. In the second part of the study we investigate the phenomenon of coexisting stable states, using the aspect ratio 2, Prandtl number 6.7 and either perfectly insulating or perfectly conducting sidewalls. Varying Rayleigh number and initial conditions, we obtain various convective patterns for the same Rayleigh number. We show also a preliminary bifurcation diagram containing stable branches. The results for perfectly insulating sidewalls are in good agreement with experiments.