Pulsar timing arrays are galactic-scale low-frequency gravitational wave observatories sensitive to the nanohertz frequency band. The primary source of gravitational radiation in this regime is expected to be a stochastic background, formed from the cosmic population of supermassive black hole binaries. In this talk, I will discuss the current state-of-the-art detection approaches to searching for a gravitational wave background in pulsar timing data and present the results obtained by analyzing the 12.5-year data release from the North American Observatory for Gravitational Waves (NANOGrav). Our analysis shows a strong preference for a stochastic process with a common amplitude in all pulsars. However, the evidence for a spatially correlated process is only slightly higher. Spatial correlations are considered the “smoking gun” of gravitational wave background detection, thus the analysis of this data set remains inconclusive. We present a suite of statistical checks to evaluate the significance of these detection metrics and discuss the expected growth in future data sets, assuming the signal is indeed an astrophysical background.