Tropical geometry can be viewed as an efficient degeneration technique in algebraic geometry, with important applications for instance in enumerative geometry. To make good use of it, it can be necessary to focus on faithful tropicalizations. Modifications provide a concrete tool to construct faithful tropicalizations. We discuss the use and effect of modifications in the case of curves of genus one, two and three. This talk is based on joint work with Maria Angelica Cueto and Ralph Morisson.