This note deals with the optimal tuning of the free parameters of the so-called Tuned Mass Dampers (TMDs), devices which are conceived to be attached to structural systems in order to reduce the vibrations induced by external actions. First, the basic concepts of classical Den Hartog’s tuning are briefly reviewed and analysed systematically in the general context of damping of the main system and of the TMD device, both for harmonic excitation at the principal mass or at the support. Analytical results are summarised in table form, whereby the different explicit expressions of the normalised amplitude x1 /xst of the primary mass oscillation are given as a function of the various system parameters. Some 2D and 3D maps are also provided to show the role played by the tuning parameters. Optimal tuning is then investigated numerically on a SDOF + TMD system by a minimax procedure. Comparisons to the fundamental tuning by Den Hartog are provided and the effect of structural damp-ing is explored. Useful abacuses are also compiled, for easier access to the best tuning. The dynamical response of a 10-storey shear-type building with a TMD added on top, subjected to El Centro input ground motion is then evaluated numerically. Results show and quantify the reduction of the top-floor displacement after the insertion of the TMD.