  We study isometric maps between Teichm\"uller spaces and bounded symmetric
domains in their intrinsic Kobayashi metric. From a complex analytic
perspective, these two important classes of geometric spaces have several
features in common but also exhibit many differences. The focus here is on
recent results proved by the author; we give a list of open questions at the
end. Paper based on plenary talk presented to the 6th Ahlfors-Bers Colloquium
at Yale, 23-26 October 2014.
