For a fixed group G, we study the model theory of actions of G by field automorphisms. The main question here is to characterize the class of groups G for which the theory of such actions has a model companion (a first-order theory of "large" actions). In my talk, I will discuss several classes of groups G in this context. The case of finite groups is joint work with Daniel Hoffmann ("Existentially closed fields with finite group actions", Journal of Mathematical Logic, (1) 18 (2018), 1850003). The case of finitely generated virtually free groups is joint work with à zlem Beyarslan ("Model theory of fields with virtually free group actions", Proc. London Math. Soc., (2) 118 (2019), 221-256). The case of commutative torsion groups is joint work with à zlem Beyarslan