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.