Abstract: Evidence of a gravitational wave (GW) signal has emerged in pulsar timing array (PTA) data, opening a new window into the nanohertz GW Universe. We explore the physics of GW signals that may explain the data, with a focus on GW backgrounds (GWBs) considering both astrophysical and cosmological origins. We describe how: An astrophysical nanohertz GWB emerges as the superposition of individual signals from inspiraling massive black hole binaries. Environment coupling, eccentricity, and sparse sampling cause great uncertainty in the theoretical prediction of the supermassive black hole signal but also offer a way to determine the origin of the signal. PTA data offer unprecedented opportunities to constrain high-energy physics beyond the Standard Model by probing early Universe GWBs that originated during or after inflation. Different early Universe GWBs, typically created by nonlinear and out-of-equilibrium dynamics, can explain the PTA data (e.g., those from inflation scenarios, first-order phase transitions, or topological defects). The PTA detection of GWs opens a new window to explore the Universe, with profound implications for astrophysics and particle physics (probing, e.g., the equation of state of the early Universe, the origin of cosmological perturbations, the nature of dark matter, or whether exotic objects like primordial black holes or cosmic strings exist).