  This review summarizes and amplifies on recent investigations of coupled
quantum dynamical systems in the short wavelength limit. We formulate and
attempt to answer three fundamental questions: (i) What drives a dynamical
quantum system to behave classically ? (ii) What determines the rate at which
two coupled quantum--mechanical systems become entangled ? (iii) How does
irreversibility occur in quantum systems with few degrees of freedom ? We embed
these three questions in the broader context of the quantum--classical
correspondence, which motivates the use of short--wavelength approximations to
quantum mechanics such as the trajectory-based semiclassical methods and random
matrix theory. Doing so, we propose a novel investigative procedure towards
decoherence and the emergence of classicality out of quantumness in dynamical
systems coupled to external degrees of freedom. We reproduce known results
derived using master equation or Lindblad approaches but also generate novel
ones. In particular we show how local exponential instability also affects the
temporal evolution of quantum chaotic dynamical systems. We extensively rely on
numerical experiments to illustrate our findings and briefly comment on
possible extensions to more complex problems involving environments with $n \gg
1$ interacting dynamical systems, going beyond the uncoupled harmonic
oscillator model of Caldeira and Leggett.
