We consider liquid in a circular cylinder that undergoes nonlinear Marangoni insta- bility. The upper free surface of the liquid is taken to have large-enough surface tension that surface deflections are neglected. The side walls are adiabatic and impenetrable, and for mathematical simplicity the liquid is allowed to slip on the side walls. The linearized stability theory for heating from below gives the critical Marangoni number Mc as a function of cylinder dimensions, surface-cooling condition and Rayleigh number. The steady nonlinear convective states near Mc are calculated using an asymptotic theory, and the stability of these states is examined. At simple eigenvalues Mc the finite-amplitude states are determined. We find th at the Prandtl number of the liquid influences the stability of axisymmetric states, distinguishing upflow at the centre from downflow. Near those aspect ratios corresponding to double eigenvalues Me, where two convective states of linear theory are equally likely, the nonlinear theory predicts sequences of transitions from one steady convective state to another as the Marangoni number is increased. These transitions are determined and discussed in detail. Time-periodic convection is possible in certain cases.