I will explain some results and conjectures relating the representation theory of simple affine vertex algebras to the geometry of homogeneous elliptic affine Springer fibres associated with the Langlands dual affine Lie algebra. I will also explain generalizations of this relation to chiral algebras associated with an n-punctured Riemann surface of genus g, constructed by T. Arakawa, and the geometry of the Hitchin moduli space on the same Riemann surface.