In the present work the problem of optimizing the geometry of internally finned tubes in order to enhance the heat transfer under laminar flow conditions is studied. The velocity and temperature distributions on the finned tube cross-section are determined with the help of a finite element model, and a global heat transfer coefficient is calculated. A polynomial lateral profile is proposed for the fins and the geometry is optimized in order to make the heat transferred per unit of tube length or surface as high as possible for a given weight and for a given hydraulic resistance. Finally, the optimum fin profiles obtained by means of a genetic algorithm are shown for different situations.