Historically, the study of the (outer) automorphism group of a given group has interested group-theorists, topologists and geometers, and consequently it is also of great importance in the Lie algebra theory.

In this talk, we will compare the Schreier and Zassenhaus conjectures on the solvability of Out(G) (resp. Out(L)), the group of outer automorphisms (resp. the Lie algebra of outer derivations) of a finite simple group G (resp. a finite-dimensional simple Lie algebra L). While the former is known to be true as a consequence of the classification of finite simple groups, the latter is false over fields of small characteristic p=2,3. Therefore, we will devote a part of the talk to give a quick overview of the simple Lie algebras of classical and Cartan type over fields of positive characteristic. Finally we will finish by presenting a new family of counterexamples to the Zassenhaus conjecture over fields of characteristic p=3, as well as commenting some advances for p=2.