such a unification beyond the SM is to explain the quantification of the electric charge, that is to explain why 3 Q (quark) = Q(electron). Various Lie group candidates are SU ( n + 1), SO (4 n + 2) families and E$_{6}$ , when imposing that the groups must contain anomaly free complex representations to accommodate the known fermions of the SM [430]. The smallest and most studied of these candidates are SU (5), SO (10), and E$_{6}$ . For reviews, see [38, 431– 434] and more specifically [39] for proton decay, and [430] for group theory and useful tables on symmetry breaking. Although the idea of gauge unification is very appealing, but this concept has additional constraints. Clearly the GUT group must contain G$_{GUT}$ ⊃ G$_{SM}$ and must break sponta- neously into its subgroups and finally G$_{SM}$ at some high scales. During this breaking the extra gauge bosons of G$_{GUT}$ , which carry quantum numbers of several groups and G$_{SM}$ , acquire a superheavy mass, M$_{GUT}$ , which prevent from a too rapid decay of the nucleons. The choice of G$_{GUT}$ and the presence of SUSY affect the predictions for proton lifetime of a given model as the scale of SUSY breaking and the particles involved in Feynman graphs modify the results. Finally, the electroweak precision measurements require the particles beyond the SM to be heavy enough, for instance the famous doublet-triplet splitting of the SM higgs from the additional Higgs fields (see the SU (5) example below). 1. SU (5) and SO (10) GUT The most studied candidate is based on SU (5) whose generators are represented by five- by-five traceless, Hermitian matrices. SU (3) × SU (2) × U (1) is one of its maximal subgroup (of same rank) together with SU (4) × U (1). Let us highlight the main features of the fermionic, bosonic and the Higgs sectors. If the gauge bosons necessarily belong to the adjoint representation of dimension 24, the fermions and the Higgs can be accommodated in any representations that lead to the right phenomenology. The smallest representations for SU (5) are the fundamental 5 , 10 , 15 , 24 , ... . First, the 15 fermions can be put in a fundamental ¯ 5 and 10 , a anti-symmetric traceless 5 × 5 matrix [435]. The decomposition of these representations under SU (3)$_{c}$ × SU (2)$_{L}$ lead to the quantum numbers of the fermions under the SM: ¯ 5 = ( ¯ 3 , 1 ) + ( 1 , 2 ) and 10 = ( 3 , 2 ) + ( ¯ 3 , 1 ) + ( 1 , 1 ) . (148) 66