  While non-abelian groups are undoubtedly the cornerstone of Grand Unified
Theories (GUTs), phenomenology shows that the role of abelian and discrete
symmetries is equally important in model building. The latter are the
appropriate tool to suppress undesired proton decay operators and various
flavour violating interactions, to generate a hierarchical fermion mass
spectrum, etc. In F-theory, GUT symmetries are linked to the singularities of
the elliptically fibred K3 manifolds; they are of ADE type and have been
extensively discussed in recent literature. In this context, abelian and
discrete symmetries usually arise either as a subgroup of the non-abelian
symmetry or from a non-trivial Mordell-Weil group associated to rational
sections of the elliptic fibration. In this note we give a short overview of
the current status and focus in models with rank-one Mordell-Weil group.
