  The strong beams of high-frequency gravitational waves (GW) emitted by cusps
and kinks of cosmic strings are studied in detail. As a consequence of these
beams, the stochastic ensemble of GW's generated by a cosmological network of
oscillating loops is strongly non Gaussian, and includes occasional sharp
bursts that stand above the ``confusion'' GW noise made of many smaller
overlapping bursts. Even if only 10% of all string loops have cusps these
bursts might be detectable by the planned GW detectors LIGO/VIRGO and LISA for
string tensions as small as $G \mu \sim 10^{-13}$. In the implausible case
where the average cusp number per loop oscillation is extremely small, the
smaller bursts emitted by the ubiquitous kinks will be detectable by LISA for
string tensions as small as $G \mu \sim 10^{-12}$. We show that the strongly
non Gaussian nature of the stochastic GW's generated by strings modifies the
usual derivation of constraints on $G \mu$ from pulsar timing experiments. In
particular the usually considered ``rms GW background'' is, when $G \mu \gaq
10^{-7}$, an overestimate of the more relevant confusion GW noise because it
includes rare, intense bursts. The consideration of the confusion GW noise
suggests that a Grand Unified Theory (GUT) value $ G \mu \sim 10^{-6}$ is
compatible with existing pulsar data, and that a modest improvement in pulsar
timing accuracy could detect the confusion noise coming from a network of cuspy
string loops down to $ G \mu \sim 10^{-11}$. The GW bursts discussed here might
be accompanied by Gamma Ray Bursts.
