Pulsar timing array (PTA) collaborations in North America, Australia, and Europe, have been exploiting the exquisite timing precision of millisecond pulsars over decades of observations to search for correlated timing deviations induced by gravitational waves (GWs). PTAs are sensitive to the frequency band ranging just below 1 nanohertz to a few tens of microhertz. The discovery space of this band is potentially rich with populations of inspiraling supermassive black-holes binaries, decaying cosmic string networks, relic postinflation GWs, and even non-GW imprints of axionic dark matter.This article aims to provide an understanding of the exciting open science questions in cosmology, galaxy evolution, and fundamental physics that will be addressed by the detection and study of GWs through PTAs. The focus of the article is on providing an understanding of the mechanisms by which PTAs can address specific questions in these fields, and to outline some of the subtleties and difficulties in each case. The material included is weighted most heavily towards the questions which we expect will be answered in the near-term with PTAs; however, we have made efforts to include most currently anticipated applications of nanohertz GWs.PACS numbers: § Note, a more precise remark is that the Earth and pulsar terms are not correlated as long as two pulsars are separated by many gravitational wavelengths, that is to say that f L 1, where f is the GW frequency and L is distance to the pulsar. This assumption is called the short-wavelength approximation (e. g. Mingarelli & Mingarelli 2018).