  We analyze the Crab pulsar at ten frequencies from 0.43 to 8.8 GHz using data
obtained at the Arecibo Observatory. Giant pulses occur only in the main and
interpulse components manifest from radio frequencies to gamma-ray energies.
Individual giant pulses reach brightness temperatures of at least $10^{32}$K in
our data, which do not resolve the narrowest pulses, and are known to reach
$10^{37}$K in nanosecond-resolution observations (Hankins et al 2003). Giant
pulses are therefore the brightest known in the observable universe and
represent an important milestone for theories of the pulsar emission mechanism
to explain. Their short durations allow them to serve as especially sensitive
probes of the Crab Nebula and the interstellar medium. We analyze frequency
structure in individual giant pulses using a scintillating,
amplitude-modulated,polarized shot-noise model. The frequency structure
associated with multipath propagation decorrelates on a time scale of 25 sec at
1.5 GHz, which requires that multipath propagation be strongly influenced by
material within the Crab Nebula. Additional frequency structure decorrelates
faster than one spin period, as would be expected from the shot-noise pattern
of nanosecond duration pulses emitted by the pulsar. Taking into account the
Crab pulsar's locality inside a bright supernova remnant, we conclude that the
brightest pulse in a typical 1-hour observation would be most easily detectable
in our lowest frequency band (0.43 GHz) to a distance of 1.6 Mpc. We also
discuss the detection of such pulses using future instruments such as LOFAR and
the SKA.
