Gromov-Witten invariants are defined via intersection theory on Kontsevich's space of stable maps. The latter is very singular, with many irreducible components of different dimensions. As a consequence the invariants are difficult to compute and to interpret. In this talk I will explain how to construct smooth modular compactifications of maps to projective space in low genus. The construction will involve tropical techniques and the study of curve singularities worst than nodal. This is based on a joint work with L. Battistella.