  We have begun an exciting era for gravitational wave detection, as several
world-leading experiments are breaching the threshold of anticipated signal
strengths. Pulsar timing arrays (PTAs) are pan-Galactic gravitational wave
detectors that are already cutting into the expected strength of gravitational
waves from cosmic strings and binary supermassive black holes in the
nHz-$\mu$Hz gravitational wave band. These limits are leading to constraints on
the evolutionary state of the Universe. Here, we provide a broad review of this
field, from how pulsars are used as tools for detection, to astrophysical
sources of uncertainty in the signals PTAs aim to see, to the primary current
challenge areas for PTA work. This review aims to provide an up-to-date
reference point for new parties interested in the field of gravitational wave
detection via pulsar timing.
