In this talk we will show recent development around an important (and often) difficult question regarding Hopf algebras arising in combinatorics. Namely, we will discuss antipode formulas of well-known Hopf algebras and how they have been extended to certain families of combinatorial objects. We will see different perspectives to attack the antipode problem. Also, we will discuss, if time permits, antipode formulas arising in particular Hopf monoids. This talk is work in progress with N. Bergeron, and previous work with B. Sagan, J. Hallam, J. Machacek.