Classical Gromov-Witten invariants of the complex projective plane count the number of curves of a fixed genus and degree passing through a generic configuration of points. Over the real numbers, the naive count of such curves is not invariant under deformations, but a signed count yields a well-defined invariant. In $\mathbb{A}^1$-enumerative geometry, one seeks refinements of such invariants over arbitrary fields, capturing richer arithmetic and geometric structures. Motivic Gromov-Witten invariants provide a quadratic enrichment of curve counting, encoding information beyond the classical and real settings. This perspective has led to striking results, including refinements of the real invariants and new insights into the structure of enumerative invariants over general fields. In this talk, we will introduce the framework of motivic Gromov-Witten invariants, and discuss recent developments in computational results via tropical geometry.
