  We provide a set of theoretical constraints on models in which the Standard
Model field content is extended by vector-like fermions and in some cases also
by a real scalar singlet. Our approach is based on the study of electroweak
vacuum stability, perturbativity of model couplings and gauge couplings
unification with the use of renormalization group equations. We show that
careful analysis of these issues leads to strong constraints on the parameter
space of the considered models. This, in turn, has important implications for
phenomenology, as we show using examples of the electroweak phase transition
and the double Higgs boson production.
