The strong beams of high-frequency gravitational waves (GW's) emitted by cu
sps 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 ne
twork of oscillating loops is strongly non-Gaussian, and includes occasiona
l sharp bursts that stand above the "confusion" GW noise made of many small
er overlapping bursts. Even if only 10% of all string loops have cusps thes
e bursts might be detectable by the planned GW detectors Laser Interferomet
ric Gravitation Observatory (LIGO)-VIRGO and Laser Interference Space Anten
na (LISA) for string tensions as small as G mu similar to 10(-13). In the i
mplausible case where the average cusp number per loop oscillation is extre
mely small, the smaller bursts emitted by the ubiquitous kinks will be dete
ctable by LISA for string tensions as small as G mu similar to 10(-12). We
show that the strongly non-Gaussian nature of the stochastic GWs generated
by strings modifies the usual derivation of constraints on G mu from pulsar
timing experiments. In particular the Usually considered "rms GW backgroun
d" is, when G mu less than or similar to 10(-7), an overestimate of the mor
e relevant confusion GW noise because it includes rare, intense bursts. The
consideration of the confusion GW noise suggests that a grand unified theo
ry value G mu similar to 10(-6) is compatible with existing pulsar data, an
d that a modest improvement in pulsar timing accuracy could detect the conf
usion noise coming from a network of cuspy string loops down to G mu simila
r to 10(-11). The GW bursts discussed here might be accompanied by gamma ra
y bursts.