  Quantum chaos of many-body systems has been swiftly developing into a vibrant
research area at the interface between various disciplines, ranging from
statistical physics to condensed matter to quantum information and to
cosmology. In quantum systems with a classical limit, advanced semiclassical
methods provide the crucial link between classically chaotic dynamics and
corresponding universal features at the quantum level. Recently,
single-particle techniques dealing with ergodic wave interference in the usual
semiclassical limit $\hbar \rightarrow 0$ have begun to be transformed into the
field theoretical domain of N-particle systems in the analogous semiclassical
limit $\hbar_{eff} = 1/N \rightarrow 0$, thereby accounting for genuine
many-body quantum interference. This semiclassical many-body theory provides a
unified framework for understanding random-matrix correlations of both
single-particle and many-body quantum chaotic systems. Certain braided bundles
of classical orbits, and of mean field modes, govern interference,
respectively, and provide the key to the foundation of universality. Case
studies presented include a many-body version of Gutzwiller's trace formula for
the spectral density and out-of-time-order correlators along with brief remarks
on where further progress may be forthcoming.
