From Veltman’s Conditions to Finite Unification

First we review Veltman's suggestion to attack the naturalness problem in the Standard model by requiring absence of quadratic divergences and the resulting mass formula. Then we emphasise the influence of Veltman's suggestion in strengthening the belief that supersymmetry is the natural playground for solving the problem of quadratic divergences. Going further, we recall few sporadic suggestions concerning the cancellation of the logarithmic divergences too, which in the framework of supersymmetry has led to the construction of all-loop Finite Theories with the use of the idea of reduction of couplings. Eventually, we concentrate on a specific Finite Unified Theory and its successful predictions for the top and Higgs mass, among others, and the prospects of its final justification in future collider searches.