Gravitational waves are long-sought predictions of general relativity that have vast potential to inform our understanding of gravity and extreme astrophysical systems. By precisely timing multiple millisecond pulsars over time scales of decades, pulsar timing arrays are capable of detecting nanohertz gravitational waves from, among other things, supermassive black hole binaries and cosmic strings. I will describe how gravitational waves influence the times of arrival of pulses from pulsars and discuss several types of gravitational wave signals pulsar timing arrays are working to detect. I will highlight recent results from the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) and its international collaborators. I will discuss the near-future prospects for pulsar timing array efforts, how those prospects rely on the continuation of existing radio telescopes (such as Arecibo and The Green Bank Telescope), and how future radio telescopes (such as MeerKAT and FAST) will contribute.