  This is the write-up of a talk given in honour of Prof. Ihara's 80th Birthday
conference in Kyoto in 2018. After briefly reviewing the work of Ihara on the
projective line minus 3 points, I outline the main ideas in the proof of the
Deligne-Ihara conjecture and provide an update on recent progress in this area
and raise some new questions.
  The second part of the talk outlines the main features of the corresponding
theory in genus one, i.e., on the moduli stack of elliptic curves. It contains
many concrete and elementary examples to illustrate the main features of the
theory and to make it more widely accessible. In particular, the case of
non-abelian cocycles associated to a pair of modular forms of level one is
discussed in some detail. It also contains some new conjectures which have not
previously appeared in print.
