Abstract: Grothendieck defined the "torsion index" of a compact Lie group, which in a sense measures its topological complexity. We compute the torsion index for all simply connected groups as well as all adjoint groups. Many cases were already known, especially through the work of Tits. Among the new cases are the groups Spin(n) for all n and the biggest exceptional group, E_8. These are purely topological calculations, but (as Grothendieck showed) knowing the torsion index of G gives information about the classification of G-torsors over an arbitrary field. In particular, we get the optimal estimate for the degree of the field extension needed to trivialize arbitrary E_8-torsors over afrbitrary fields. Finally, we describe some beautiful recent counterexamples by Florence and Parimala.