In a lecture I have heard that the antipode of a finite-dimensional Hopf algebra is conjectured to be finite, and that this has only been proven in characteristic $0$ by Larson and Radford in 1988. I couldn't find such a result (only stronger statements that the antipode has order $2$ or $4$ in some nice situations). But I have found the older paper
D. E. Radford, The antipode of a finite-dimensional Hopf algebra over a field has finite order. Bull. Amer. Math. Soc. 81 (1975), no. 6, 1103--1105
which gives a proof sketch (which I cannot verify) that the conjecture is true in general. What is the actual status of the conjecture?
Related (weaker) question: Is the antipode of a finite-dimensional Hopf algebra always invertible? If so, is there an easy proof for this statement, which one can understand without much background? Takeuchi has constructed an infinite-dimensional Hopf algebra whose antipode is not invertible.