The counts of algebraic curves in projective space (and other
toric varieties) has been intensely studied for over a century.
The subject saw a major advance in the 1990s, due to
groundbreaking work of Kontsevich in the 1990s. Shortly after,
considerations from high energy physics led to an entirely
combinatorial approach to these curve counts, via piecewise
linear embeddings of graphs, pioneered by Mikhalkin. I will
give an introduction to the surrounding ideas, outlining new
results and new proofs that the theory enables. Time permitting
I will discuss generalizations, difficulties, and future
directions for the subject.