Theory challenges at future lepton colliders

Abstract: High energy, high luminosity, future lepton colliders, circular or linear, may possibly give us hint about fundamental laws of Nature governing at very short distances and very short time intervals, the same which have brought our Universe to live. Currently considered projects are on one hand linear electron-positron colliders, which offer higher energy and lower beam intensities and on the other hand circular electron-positron colliders, limited in energy but offering tremendous interaction rates. On the far future horizon, muon circular colliders are the only viable projects which can explore $>10$TeV teritory of the lepton colliders. Experiments in all these future colliders will require theoretical calculations, mainly of Standard Model processes (including QED), at the precision level one or even two orders better than available today. After briefly characterization of theory puzzles in the fundamental interactions we shall overview main challenges in the precision calculations of the Standard Model effects, which have to be removed from data, in order to reveal traces of new unexpected phenomena.