Pulsar Timing Arrays (PTAs) are currently the only way to search for gravitational radiation in the nanohertz band. Sources of interest include gravitational wave (GW) backgrounds generated by supermassive black hole binaries (SMBHBs) and processes in the early universe such as relic GWs and cosmic strings. Limits on the GW background continue to improve, and searches of increasing sensitivity are ongoing. Here I will discuss recent limits on the stochastic GW background, how we can characterize the stochastic GW background from SMBHBs on any angular scale to search for anisotropy, and how environmental interactions with SMBHBs affect the GW strain spectrum, and thus become observable. I then discuss how PTAs can place limits on the amplitude of the relic GW background and may eventually be a valuable tool for cosmology by providing independent constraints on the tensor-to-scalar ratio “r”, and the spectral index of the tensor fluctuation spectrum “n_t”. Finally, I present recent NANOGrav limits on cosmic string tension, which are a factor of 4 more constraining than the Planck (2014) results.