**Title:**Enumerative geometry beyond

**C**

**Abstract:**There are 27 lines on a cubic surface, 8 circles tangent to 3 given circles, and a single line through 2 points. For millennia, mathematicians have sought to count solutions to geometric problems. However, most theorems in enumerative geometry require one to work over an algebraically closed field (such as the complex numbers). Using the 27 lines on a cubic surface as a case study, I will discuss some history about enumerative geometry over non-algebraically closed fields. Time permitting, I will describe some recent results in enriched enumerative geometry, where one uses tools from motivic homotopy theory to give quadratic form-valued counts.