  The ground states of all even-even nuclei have angular momentum, $I$, equal
to zero, I=0, and positive parity, $\pi=+$. This feature was believed to be a
consequence of the attractive short-range interaction between nucleons.
However, in the presence of two-body random interactions, the predominance of
$I^{\pi}=0^+$ ground states (0 g.s.) was found to be robust both for bosons and
for an even number of fermions. For simple systems, such as $d$ bosons, $sp$
bosons, $sd$ bosons, and a few fermions in single-$j$ shells for small $j$,
there are a few approaches to predict and/or explain spin $I$ ground state ($I$
g.s.) probabilities. An empirical approach to predict $I$ g.s. probabilities is
available for general cases, such as fermions in a single-$j$ ($j>7/2$) or
many-$j$ shells and various boson systems, but a more fundamental understanding
of the robustness of 0 g.s. dominance is still out of reach. Further
interesting results are also reviewed concerning other robust phenomena of
many-body systems in the presence of random two-body interactions, such as the
odd-even staggering of binding energies, generic collectivity, the behavior of
average energies, correlations, and regularities of many-body systems
interacting by a displaced two-body random ensemble.
