This cycle of talks wants to highlight how ideas from tropical geometry have contributed not only to the solution, but also to the development of enumerative geometric problems regarding moduli spaces of curves, and maps from curves to curves. We will spend a little of time reviewing the origins of this story, i.e. the development of tropical Hurwitz numbers as combinatorial analogues for the classical Hurwitz numbers. We will discuss a more recent interpretation that views tropical Hurwitz numbers as the natural computation for the intersection number of the double ramification cycle with an element of the log Chow ring of the moduli space of curves (called in this case the branch polynomial, as it is presented as a piecewise polynomial function on the moduli spaces of tropical curves) which is determined by the tropical moduli space of covers of the projective line. We will see that from the tropical perspective analogous piecewise polynomial functions may be associated to $k$-DR cycles (cycles arising from spaces of twisted pluri-differentials), thus giving rise to $k$-analogues of Hurwitz numbers (called leaky Hurwitz numbers) that enjoy many of the algebro-combinatorial properties of Hurwitz numbers - such as piecewise polynomiality and wall crossings.
We will present some work in progress which intends to incorporate descendants into these pictures. Tropical algorithms are developed that give rise to some intriguingly simple formulas in the case when one point is fully ramified. The material presented is based on many years of joint work with several people, including Paul Johnson, Hannah Markwig, Dhruv Ranganathan and Johannes Schmitt.