By interpreting $1$ as the unique complex quadratic form $z \mapsto z^2$, some classical enumerations (i.e., with values in $\mathbb N$) acquire meaning when the field of complex numbers is replaced with an arbitrary field $k$. The result of the enumeration is then a quadratic form over $k$ rather than an integer. This talk will focus on such enumeration for rational curves in surfaces, that are, roughly speaking, curves admitting a parameterization $k\mapsto k^2$. I will explain how this quadratic count is defined, and how these quadratic invariants are related to enumeration of complex and real curves (i.e., to Gromov-Witten invariants and Welschinger invariants, respectively.)
