After quickly discussing classical uniformisation results for Kähler-Einstein projective manifolds with ample, torsion or anti-ample canonical bundle, I will explain some sheaf-theoretic techniques that allow to characterise projective spaces among Fano manifolds satifying a certain algebrogeometric stability condition. While the discussion will be aimed towards graduate students, towards the end I will shortly discuss how the techniques generalise to the case of varieties with singularities as they appear in the minimal model program and with nef anti-canonical bundle.