The problem of counting complex algebraic curves of a given degree that pass through a collection of points in the plane dates back to the 19th century. The count does not depend on the choice of the points. More recently, the counting problem was extended to real algebraic curves. In this case, the naive count depends on the choice of points, but an appropriate signed count is invariant. I will discuss joint work with Kass, Levine and Wickelgren, on curve counting over more general fields. The invariant count is not a number but rather a quadratic form over the given field. The complex and real counts can be recovered as the rank and signature of the quadratic form respectively. We follow a topological approach, based on motivic homotopy theory, in the spirit of Gromov-Witten theory.