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.