We consider several first passage problems for stable processes, giving explicit formulas for hitting distributions, hitting probabilities and potentials of stable processes killed at first passage. Our principal tools are the Lamperti representation of positive self-similar Markov processes and the Wiener–Hopf factorisation of Lévy processes. As part of the proof apparatus, we introduce a new class of Lévy processes with explicit Wiener–Hopf factorisation, which appear repeatedly in Lamperti representations derived from stable processes. We also apply the Lamperti–Kiu representation of real self-similar Markov processes and obtain results on the exponential functional of Markov additive processes, in order to find the law of the first time at which a stable process reaches the origin.