Gravitational-wave (GW) astronomy with ground-based interferometric detectors is rapidly entering its golden age. The first three observing runs of the LIGO-Virgo-KAGRA collaboration have detected and catalogued about 90 GW signals from compact binary coalescences (CBCs), and the ongoing fourth observing run, started on 23rd May 2023, is expected to detect a CBC signal every two to three days. Among the variety of GW signals that could be detected there is the stochastic gravitational-wave background (SGWB), which is expected to arise from the superposition of all the GW signals produced by sources that cannot be individually detectable and/or are unresolvable. An SGWB can be generated by a plethora of phenomena, either of astrophysical or cosmological origin, and its spectrum may span a broad range of frequencies. Detecting an SGWB would be extraordinarily valuable. On the one hand, the discovery of an astrophysical SGWB would allow for gaining a noticeable amount of information about astrophysical populations otherwise inaccessible to electromagnetic astronomy and about the cosmos stellar history. On the other hand, the discovery of a cosmological SGWB would enable the direct assessment of inflation-like and other cosmological scenarios and the probe of physical laws at extremely high energies beyond the reach of particle physics colliders. The search for different kinds of stochastic gravitational-wave backgrounds in the frequency range of LIGO and Virgo detectors using the data from the first three LVK observing runs is the main topic of this doctoral dissertation.