In these lectures we shall explore the boundary region between algebraic and differential geometry for spaces possessing non-abelian symmetry. In algebraic geometry, the presence of a non-abelian reductive group acting on a variety gives rise to certain canonical equivariant degenerations of the variety. Moreover, the symmetry provides such control over the degenerations so as to make transitions between algebra-geometric and differential notions, which are often very difficult to handle, actually tractable. Motivating examples of this phenomenon that will be discussed include the switch between Kahler and real polarisations in geometric quantisation, or the link between K-stability and existence of Kahler-Einstein metrics.