Pulsar timing array experiments around the world have recently published the first evidence of nanohertz gravitational waves in the form of a stochastic gravitational wave background. This discovery opens another window on the gravitational wave universe that we can use to study a variety of astrophysical and cosmological sources, as well as perform tests of gravity. Pulsar timing arrays search for gravitational waves by looking for interpulsar correlations in the timing residuals of a collection of millisecond pulsars. In this talk, I will discuss how pulsar timing arrays work, what sources we expect to find emitting nanohertz gravitational waves and what we can learn about them, and how pulsar timing arrays can be used to perform tests of gravity.