We review the basic idea of the reduction of couplings method, both in the dimensionless and dimension 1 and 2 sectors. Then, we show the application of the method to $N=1$ supersymmetric GUTs, and in particular to the construction of finite theories. We present the results for two phenomenologically viable finite models, an all-loop finite $SU(5)$ SUSY GUT, and a two-loop finite $SU(3)^3$ one. For each model we select three representative benchmark scenarios.
In both models, the supersymmetric spectrum lies beyond the reach of the 14 TeV HL-LHC. For the $SU(5)$ model, the lower parts of the parameter space will be in reach of the FCC-hh, although the heavier part will be unobservable. For the two-loop finite $SU(3)^3$ model, larger parts of the spectrum would be accessible at the FCC-hh, although the highest possible masses would escape the searches.