  Simple semitoric systems were classified about ten years ago in terms of a
collection of invariants, essentially given by a convex polygon with some
marked points corresponding to focus-focus singularities. Each marked point is
endowed with labels which are symplectic invariants of the system. We will
review the construction of these invariants, and explain how they have been
generalized or applied in different contexts. One of these applications
concerns quantum integrable systems and the corresponding inverse problem,
which asks how much information of the associated classical system can be found
in the spectrum. An approach to this problem has been to try to compute
invariants in the spectrum. We will explain how this has been recently achieved
for some of the invariants of semitoric systems, and discuss an open question
in this direction.
