enumerative geometry is a branch of geometry devoted to counting geometric objects. for example, one could ask: how many lines are there passing through two points? (easy, that's one line.) or one could ask: given five lines in the plane, how many conic sections are there tangent to all five lines? (harder, but the answer is still one.) given a surface defined by a cubic equation (say $x^3+y^3+z^3=1$), how many straight lines are contained in the surface? (this was determined in the mid-19th century, and the answer is 27.) even harder, given a three-dimensional object defined by an equation like $x^5+y^5+z^5+w^5=1$, how many plane conic sections are contained in this object? (much harder, the answer is 609,250.) i will give some examples and techniques, and explain the history of how the field of enumerative geometry had a rebirth when string theory started making predictions about answers to such questions.