Summary: In this short survey paper we shall consider, in particular, indefinite integrals of differentials on algebraic curves, trying to express them in elementary terms. This is an old-fashioned issue, for which Liouville gave an explicit criterion that may be considered a primordial example of differential algebra. Before presenting some connections with more recent topics, we shall start with an overview of the classical facts, recalling some criteria for elementary integration and relating this with issues of torsion in abelian varieties. Then we shall turn to differentials in 1-parameter algebraic families, asking for which values of the parameter we can have an elementary integral. (This had been considered already in the 80s by J. Davenport.) The mentioned torsion issues provide a connection of this with a conjecture of R. Pink in the realm of the so-called unlikely intersections. In joint work in collaboration with David Masser (still partly in progress), we have proved finiteness of the set of relevant values, under suitable necessary conditions. Here we shall give a brief account of the whole context, pointing out at the end possible links with other problems.
For the entire collection see [Zbl 1314.00104].