  This is a survey article written for the Jahresberichte der DMV.
  Tropical geometry can be viewed as an efficient combinatorial tool to study
degenerations in algebraic geometry. Abstract tropical curves are essentially
metric graphs, and covers of tropical curves maps between metric graphs
satisfying certain conditions. In this short survey, we offer an introduction
to the combinatorial theory of abstract tropical curves and covers of curves,
and their moduli spaces, and we showcase three results demonstrating how this
theory can be applied in algebraic geometry.
