  Efforts to detect gravitational waves by timing an array of pulsars have
focused traditionally on stationary gravitational waves: e.g., stochastic or
periodic signals. Gravitational wave bursts --- signals whose duration is much
shorter than the observation period --- will also arise in the pulsar timing
array waveband. Sources that give rise to detectable bursts include the
formation or coalescence of supermassive black holes (SMBHs), the periapsis
passage of compact objects in highly elliptic or unbound orbits about a SMBH,
or cusps on cosmic strings. Here we describe how pulsar timing array data may
be analyzed to detect and characterize these bursts. Our analysis addresses, in
a mutually consistent manner, a hierarchy of three questions: \emph{i}) What
are the odds that a dataset includes the signal from a gravitational wave
burst? \emph{ii}) Assuming the presence of a burst, what is the direction to
its source? and \emph{iii}) Assuming the burst propagation direction, what is
the burst waveform's time dependence in each of its polarization states?
Applying our analysis to synthetic data sets we find that we can \emph{detect}
gravitational waves even when the radiation is too weak to either localize the
source of infer the waveform, and \emph{detect} and \emph{localize} sources
even when the radiation amplitude is too weak to permit the waveform to be
determined. While the context of our discussion is gravitational wave detection
via pulsar timing arrays, the analysis itself is directly applicable to
gravitational wave detection using either ground or space-based detector data.
