Enumerative geometry is a very classical branch of (complex projective) algebraic geometry, that counts solutions to geometric problems: e.g., for a cubic polynomial f(x,y,z), there are 27 lines L in space with f(p)=0 at every point in L (Cayley-Salmon, 1849).
Around 1990, the influx of ideas from string theory (in particular Gromov-Witten invariants) opened up new research directions, which have kept developinging to the present day, with continued interactions with symplectic geometry, integrable systems, and particle physics.
We will sketch a part of this story, highlighting how "abstract" notions like schemes arise naturally in very concrete problems.