  Massive black hole binary systems, with masses in the range ~10^4-10^10
\msun, are among the primary sources of gravitational waves in the frequency
window ~10^-9 Hz - 0.1 Hz. Pulsar Timing Arrays (PTAs) and the Laser
Interferometer Space Antenna (LISA) are the observational means by which we
will be able to observe gravitational radiation from these systems. We carry
out a systematic study of the generation of the stochastic gravitational-wave
background from the cosmic population of massive black hole binaries. We
consider a wide variety of assembly scenarios and we estimate the range of
signal strength in the frequency band accessible to PTAs. We show that, taking
into account the uncertainties surrounding the actual key model parameters, the
amplitude lies in the interval h_c(f = 10^-8 Hz)~5x10^-16 - 8x10^-15. The most
optimistic predictions place the signal level at a factor of ~3 below the
current sensitivity of Pulsar Timing Arrays, but within the detection range of
the complete Parkes PTA for a wide variety of models, and of the future
Square-Kilometer-Array PTA for all the models considered here. We also show
that at frequencies >10^-8 Hz the frequency dependency of the generated
background follows a power-law significantly steeper than f^-2/3, that has been
considered so far. Finally we show that LISA observations of individual
resolvable massive black hole binaries are complementary and orthogonal to PTA
observations of a stochastic background from the whole population in the
Universe. In fact, the detection of gravitational radiation in both frequency
windows will enable us to fully characterise the cosmic history of massive
black holes.
