The nucleon-antinucleon (NNBAR) annihilation process provides a fertile testing ground for microscopic hadron exchange and/or quark models. We review a variety of such approaches here, including models which treat the underlying quark-gluon degrees of freedom explicitly. A first principles calculation of NNBAR annihilation in the framework of non-perturbative quantum chromodynamics (QCD) is still beyond our reach, so we adopt a more phenomenological approach, in which we try various annihilation topologies combined with different prescriptions for the dependence of the effective quark-antiquark (QQBAR) creation/destruction operator on spin, flavor and color. The NNBAR system offers a rich ensemble of annihilation channels, whose relative branching ratios BR provide strong constraints on dynamical models. Recent experiments at LEAR show that values of BR display a significant dependence on L, the NNBAR relative orbital angular momentum, spin S and isospin I. These dynamical selection rules (DSR), i.e. the suppression of transitions allowed a priori by conservation of the quantum numbers {J, pi, C, G}, provide key signatures of the annihilation mechanism, and suggest dynamical content beyond simple statistical or SU(2)/SU(3) flavor symmetry models. We investigate to what extent the observed DSR enable us to unravel the quark dynamics of the annihilation process.