Gromov-Witten theory has developed a collection of methods for dealing with a whole series of enumerative problems in algebraic geometry, such as: How many rational curves of degree d in the plane are there passing through 3d-1 points in general position? This purely numerical version of stating the problem requires one to work over an algebraically closed field, such as the complex numbers. Over other fields, such as the real numbers, the notion of "counting" becomes more complicated. One must not only take into account the various field extensions over which solutions to the given geometric problem are defined, but there is additional "orientation" information that will affect the count. One convenient way to package all this information is to give a "count" of solutions as a quadratic form. We will explain why this is a reasonable way to study enumerative problems over an arbitrary field and detail some progress that has been made recently in carrying out this program. Part of this is a joint work with Jesse Kass, Jake Solomon and Kirsten Wickelgren.