Towards systematic exploration of multi-Higgs-doublet models

Conservative bSM models with rich scalar sector, such as multi-Higgs-doublet models, can easily accommodate the SM-like properties of the 125 GeV scalar observed at the LHC. Possessing a variety of bSM signals, they are worth investigating in fuller detail. Systematic study of these models is hampered by the highly multi-dimensional parameter space and by mathematical challenges. I outline some directions along which multi-Higgs-doublet models in the vicinity of a large discrete symmetry can be systematically explored.