In this paper, a three-dimensional finite volume model of a molten carbonate fuel cell (MCFC) is presented. The model applies a detailed electrochemical and thermal analysis to a planar MCFC stack of given geometry and assigned input flows conditions, material properties and assigned external heat losses, calculating global energy balances and reactant utilization, as well as internal temperature and chemical composition profiles, pressure profiles, polarization losses, and current density and voltage distribution over the stack. The model is calibrated on the available data for an experimented MCFC stack manufactured by Ansaldo Fuel Cells. The comparison of 2D versus 3D simulations is first discussed, showing the importance of the 3D model for an accurate analysis of real stack operation under significant heat loss conditions, but also showing the mutual influence of heat transfer between adjoining cells and evidencing the variable electrical performances between different cells of the same stack. As a counterpart, application of the 2D model, which takes advantage by much shorter calculation time while keeping a reasonable accuracy in the overall energy balances, is anyway indicated for complex cycle analysis and cycle optimization, where a complete 3D stack analysis may be limited to selected cases. Then, the model is applied to the off-design analysis of hybrid cycles based on a recuperated gas turbine arrangement where variable pressure ratios and fuel supply conditions are assumed for the fuel cell. The effects of the operating pressure and of the adoption of different fuel compositions are considered, with a cycle layout that may include natural gas reforming or is directly fed by a syngas provided by an external source, simulating different fuel feedstocks (e.g., coal gasification). Complete results are presented, in terms of fuel cell and overall cycle performances.