The paper analyzes the problem of the optimum thermal design of free and forced convection fin arrays composed of longitudinal fins with constant thickness. Two different optimization problems have been considered: the minimization of the weight for a given heat flow and the maximization of the heat flow for a given fin weight. Two different geometrical configurations of the finned array have been considered -closed array and open array- and a particular optimization method considering the number of fins of the array as discrete optimization variable, has been developed. The procedure for the optimization is given for the general case and a complete analytical solution of the problem, for the case with the tip approximated as being insulated, is developed. The paper contains several illustrative examples of the application of the proposed optimization procedure.