An investigation of laminar and fully developed mixed convection in a vertical rectangular duct is presented. The analysis refers to thermal boundary conditions such that at least one of the four duct walls is kept isothermal. The evaluation of the velocity field and of the temperature field is performed analytically. The limiting case of free convection, i.e, the case of pure buoyancy-driven flow, is discussed. Special attention is devoted to the following sets of thermal boundary conditions: (A) two facing duct walls are kept isothermal with different temperatures and the others are kept insulated; (B) two facing duct walls have a uniform wall heat flux and the others are kept isothermal with the same temperature. In both cases, the conditions for the onset of flow reversal are obtained. The friction factor is evaluated. It is shown that this parameter depends only on the duct aspect ratio in case (A), while it depends also on the ratio between the Grashof number and the Reynolds number in case (B).