As long as people have studied mathematics, they have wanted to know how many
primes there are. Getting precise answers is a notoriously difficult problem,
and the first suitable technique, due to Riemann, inspired an enormous amount
of great mathematics, the techniques and insights permeating many different
fields. In this article we will review some of the best techniques for counting
primes, centering our discussion around Riemann's seminal paper. We will go on
to discuss its limitations, and then recent efforts to replace Riemann's theory
with one that is significantly simpler.Comment: To appear in a volume dedicated to the MAA Centennial in 201
