Let G be a complex (connected) reductive group and C be a complex smooth projective curve of genus at least four. It is known that the moduli space of semistable G-bundles over C is a projective variety. The automorphism group of this variety contains the so-called tautological automorphisms: they are induced by the automorphisms of the curve C, outer automorphisms of G and tensorization by Z-torsors, where Z is the center of G. It is a natural question to ask if they generate the entire automorphism group. Kouvidakis and Pantev gave a positive answer when G=SL(n). An alternative proof has been given by Hwang and Ramanan. Later, Biswas, Gomez and Muñoz, after simplifying the proof for G=SL(n), extended the result to the symplectic group Sp(2n). All the proofs rely on the study of the singular fibers of the Hitchin fibration. In this talk, we present a recent work where, by adapting the Biswas-Gomez-Muñoz strategy, we describe the automorphism group of the connected components of the moduli space of semistable G-bundles over C, for any almost-simple group G.