This book is based on the lectures given at the Oberwolfach Seminar held in Fall 2021.Enumerative geometry and the theory of moduli spaces of curves are two cornerstones of modern algebraic geometry; the two subjects have had a significant influence on each other. In the last 15 years, discrete and combinatorial methods, systematized within tropical geometry, have begun to provide new avenues of access into these two subjects. These tropical methods find combinatorial limits in degenerations of algebraic varieties, and the resulting polyhedral combinatorics provides a fundamentally new tool to constrain and understand geometric questions. These ideas have led to new results in Brill-Noether theory, classical enumerative geometry, and the compactification of moduli spaces.These notes explore these ideas in the context of Gromov-Witten theory. Logarithmic Gromov-Witten theory lies at the heart of modern approaches to mirror symmetry, but also opens up a number of new directions in enumerative geometry of a more classical flavour. Tropical geometry forms the calculus through which calculations in this subject are carried out. The Oberwolfach seminar covered the foundational aspects of this tropical calculus, geometric aspects of the degeneration formula for Gromov-Witten invariants, and the practical nuances of working with and enumerating tropical curves. Readers of this book get an assisted entry route to the subject, focusing on examples and explicit calculations.weiterlesen