In compactifications of M-Theory, certain singularities are expected to give rise to non-abelian gauge theories. In case the compactification manifold admits an torus fibration, such singularities become degenerations of the fibre. In this framework (which is called F-Theory), the behaviour of the fibre is well-understood for elliptic surfaces, but there remain puzzles both from the perspective of mathematics and physics in higher dimensions. After an introduction, I will report on some recent progress on the interplay of the structure of degenerate fibres and the effective field theory description.