We start with a general introduction to post-Lie structures on free Lie algebras and the associated Grossman-Larson product on the universal enveloping algebra. We apply this general framework to the Ihara bracket, arising in the study of multiple zeta values. Together with Racinet's work this results in a Hopf algebra structure on formal multiple zeta values. In the second part, we introduce q-analogs of multiple zeta values and assign a Lie algebra to a depth-graded formal version of them. The Lie bracket comes from Ecalle's ari bracket on bimoulds. Again we can apply the general framework of post-Lie structures and obtain a corresponding Hopf algebra structure. If time permits, we discuss also the non depth-graded case.