  Pulsar timing arrays (PTAs) can be used to search for very low frequency
($10^{-9}$--$10^{-7}$ Hz) gravitational waves (GWs). In this paper we present a
general method for the detection and localization of single-source GWs using
PTAs. We demonstrate the effectiveness of this new method for three types of
signals: monochromatic waves as expected from individual supermassive binary
black holes in circular orbits, GWs from eccentric binaries and GW bursts. We
also test its implementation in realistic data sets that include effects such
as uneven sampling and heterogeneous data spans and measurement precision. It
is shown that our method, which works in the frequency domain, performs as well
as published time-domain methods. In particular, we find it equivalent to the
$\mathcal{F}_{e}$-statistic for monochromatic waves. We also discuss the
construction of null streams -- data streams that have null response to GWs,
and the prospect of using null streams as a consistency check in the case of
detected GW signals. Finally, we present sensitivities to individual
supermassive binary black holes in eccentric orbits. We find that a
monochromatic search that is designed for circular binaries can efficiently
detect eccentric binaries with both high and low eccentricities, while a
harmonic summing technique provides greater sensitivities only for binaries
with moderate eccentricities.
