Massive black hole binary systems, with masses in the range ∼104–1010M⊙, are among the primary sources of gravitational waves in the frequency window ∼10−9–0.1 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 regardless of the specific model of massive black hole binaries formation and evolution, the characteristic amplitude hc of the gravitational-wave stochastic background at 10−8 Hz varies by less than a factor of 2. However, taking into account the uncertainties surrounding the actual key model parameters, the amplitude lies in the interval hc(f= 10−8Hz) ≈ 5 × 10−16–8 × 10−15. The most optimistic predictions place the signal level at a factor of ≈3 below the current sensitivity of PTAs, 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 hc∝f−2/3, which has been considered so far; the value of the spectral index depends on the actual assembly scenario and provides therefore an additional opportunity to extract astrophysical information about the cosmic population of massive black holes. 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 characterize the cosmic history of massive black holes.