We are living in a sea of gravitational waves - ripples in spacetime predicted by Einstein’s General Relativity and observed directly by ground-based detectors over the last few years. Another method on the cusp of making its first observations uses objects known as pulsars, precision celestial clocks in space. I will discuss how we build our galaxy-wide observatory with pulsars and the largest radio telescopes on Earth, the current challenges and advances in our scientific understanding of pulsar timing, and the science we hope to learn about in detecting gravitational waves as we reach this new frontier of astronomy.