Building and testing models with extended Higgs sectors

Models with non-minimal Higgs sectors represent a mainstream direction in theoretical exploration of physics opportunities beyond the Standard Model. Extended scalar sectors help alleviate difficulties of the Standard Model and lead to a rich spectrum of characteristic collider signatures and astroparticle consequences. In this review, we introduce the reader to the world of extended Higgs sectors. Not pretending to exhaustively cover the entire body of literature, we walk through a selection of the most popular examples: the two- and multi-Higgs-doublet models, as well as singlet and triplet extensions. We will show how one typically builds models with extended Higgs sectors, describe the main goals and the challenges which arise on the way, and mention some methods to overcome them. We will also describe how such models can be tested, what are the key observables one focuses on, and illustrate the general strategy with a subjective selection of results.