In this talk we describe all possible isomorphisms between any pair of moduli stacks of vector bundles with fixed determinant. In particular, we prove a Torelli type theorem for these moduli stacks and describe their automorphisms. The Torelli type theorem is also extended to other moduli stacks of G-bundles. Joint works with Indranil Biswas, Tomás Gómez and Swarnava Mukhopadhyay.