The nanohertz-frequency band of gravitational waves should be awash with signals from supermassive black-hole binaries, and potentially also cosmological signatures of phase transitions, cosmic strings, and other relics of the early Universe. Pulsar-timing arrays (PTAs) like the North American Nanohertz Observatory for Gravitational waves (NANOGrav) and the International Pulsar Timing Array are poised to chart this new frontier of gravitational-wave discovery within the next several years. I will present exciting new results from recent cutting-edge searches, and discuss some milestones on the road to the next decade of PTA discovery that may include the first multi-messenger detection of black-hole binaries. With data and pulsars being constantly added to these PTAs, I will also outline new pathways to confront the emerging statistical and computational challenges within our existing Bayesian analysis frameworks. These techniques being piloted in the nanohertz-frequency band could also have crossover application to similar challenges in the LISA millihertz gravitational-wave band.