The aim of this thesis is to present and discuss some mathematical and physical problems in the theory of weak, electromagnetic and strong interactions. Our main concern is a parallel development of mathematical and physical concepts and when it is possible, an attempt to bridge the abstract mathematical formulations with physical ideas. A central role in this thesis is played by a general construction scheme, which enables us to calculate explicitly all the mathematical quantities like matrix elements, Clebsch-Gordan series, Clebsch-Gordan coefficients which are necessary for a Grand Unification model construction. In this content, we have followed two basic principles: simplicity and applicability. To meet the first principle, all the construction methods developed are based on first principles and basic concepts of the Lie algebras and its representation theory, like roots and weights. Moreover, the requirement of applicability is met with the implementation of all the algorithms into computer programs. In the physical area, we have concentrated on the problem of mass. The lepton mass spectrum us studied in a theory of weak and electromagnetic interactions, while the mass problem of the SO(10) Grand Unified theory is analysed as a direct application of our Lie group construction scheme.