Grand Unified Theories (GUTs)
Despite the popular claims, the SU(2) x U(1) Standard model of elementary particle interactions formulated by Glashow, Salam, and Weinberg at late 1960's cannot actually be viewed as a unified description of the electromagnetic and weak nuclear forces. The reason is that there still remain two distinct mathematical structures (i.e., gauge groups) describing the two sectors almost independently, with just a certain cross-over parametrized by the famous Weinberg angle. Moreover, the strong nuclear interactions (which have been understood to obey the gauge principle as well; the relevant SU(3) gauge field theory established in 1970's comes under the name of Quantum Chromodynamics or QCD) are just attached to the electroweak sector with only a relatively little theoretical overlap (provided namely by the anomaly cancellation conditions).
However, the peculiarities of the matter fermion spectrum (in particular, the family structure, quantization of the electric charge, and the anomaly cancellations) together with the indication that the couplings of the gauge interactions, subject to the renormalization group running, tend to converge at higher energies lead to speculations about the common origin of all these three classes of phenomena at a very high scale MG, typically of the order of 1016 GeV. Remarkably enough, this number is close to the scale of the new physics indicated independently by the experimental fact that at least two neutrinos do have very tiny but clearly nonzero masses. If neutrinos are Majorana particles, the effective dimension 5 operators contributing at the leading order to their masses should contain a large (cut-off) scale MN in the denominators; numerically, one typically obtains MN in the range of 1012-1014 GeV, in a remarkable proximity to MG.
This all points to a hypothesis that there might be something very interesting going on at the Peta-GeV scale. The running behavior of the couplings corresponding to the three distinct sectors of the Glashow-Weinberg-Salam (+QCD) model viewed as an effective low-scale SU(3) x SU(2) x U(1) gauge field theory is compatible with the embedding of this structure into a larger simple gauge group such as SU(5) and/or SO(10). Note that for the SU(5) case this is achievable, namely in the supersymmetric framework, in which at least some of the SUSY particles are supposed to show up around the TeV scale, which, however, is to be directly probed by the various experiments at the large hadron collider (LHC) in the near future.
In the "grand unified" theory (GUT) approach the six different multiplets of matter fermions of each generation (taking into account the right-handed neutrinos playing a central role in the seesaw mechanism) are understood as different facets of a limited number of more "elementary" objects, typically 3 (of dimensions 10, 5, and 1) in SU(5) kind of scenarios or even a single 16-dimensional one in SO(10)-like theories. It is then only the fact that we are observing the world from a point, where the apparent symmetry is considerably smaller, which makes us believe there are actually three different interactions governing the world of a multitude of distinct elementary particles. However, at least some hints on the underlying symmetry could remain visible even from our limited perspective - the unified structure might be imprinted e.g. into the correlations between the various Yukawa couplings that remain essentially unconstrained in the Standard Model. Next, the gauge interactions mediate new kind of transitions between quarks and leptons (which in GUTs occupy common multiplets) with profound implications for the nucleon stability. Last, but not least, the breakdown of the GUT-scale symmetry typically leads to a production of somewhat "exotic" objects called monopoles in the early stage of the Universe, which are being under a constant pursuit for their peculiar features, since the birth of the unification idea.
Unlike many other modern particle physics concepts, the grand unified theories generally also do obey the central physicality criterion, which, according to Karl Popper, is falsifiability rather than verifiability - though the inflationary epoch of the early Universe can reduce the current flux density of the GUT monopoles below any measurable bounds, the gauge-interaction-mediated proton decay is an almost inevitable consequence of the GUT concept. This makes this beautiful class of theories experimentally testable in a relatively near future.