We review recent results on quantum reactive scattering from a phase space perspective. The approach uses classical and quantum versions of Poincaré-Birkhoff normal form theory and the perspective of dynamical systems theory. Over the past 10 years the classical normal form theory has provided a method for realizing the phase space structures that are responsible for determining reactions in high-dimensional Hamiltonian systems. This has led to the understanding that a new (to reaction dynamics) type of phase space structure, a normally hyperbolic invariant manifold (or NHIM), is the "anchor" on which the phase space structures governing reaction dynamics are built, e.g., it is the classical analogue of the chemists notion of the "activated complex" and it is essential for the construction of a surface that divides reactants from products which has the "no-recrossing" property for trajectories and minimal flux. The quantum normal form (QNF) theory provides a method for quantizing these phase space structures through the use of the Weyl quantization procedure. We show that this approach provides a solution of the time-independent Schrödinger equation leading to a (local) S-matrix in a neighborhood of the saddle point governing the reaction. These results can be obtained for any dimensional system for which an accurate normal form can be computed, and it does not require numerical solution of the Schrödinger equation or the generation of any classical trajectories. It follows easily that the quantization of the directional flux through the dividing surface with the properties noted above is a flux operator that can be expressed in a "closed form." Moreover, from the local S-matrix we easily obtain an expression for the cumulative reaction probability (CRP), which is the essential ingredient for the computation of microcanonical reaction rates and thermal reaction rates. Significantly, the expression for the CRP can be evaluated without the need to compute classical trajectories. This is a by-product of the quantization of classical phase space structures that govern "exact" classical dynamics. The quantization of the NHIM is shown to lead to the activated complex, and the lifetimes of quantum states initialized on the NHIM correspond to the Gamov-Siegert resonances. We apply these results to the collinear nitrogen exchange reaction and a three degree-of-freedom system corresponding to an Eckart barrier coupled to two Morse oscillators. We end by describing some further challenges that are topics of current research, but where some preliminary results are known: corner-cutting tunneling, state-to-state reaction rates, the flux-flux autocorrelation function formalism, and the convergence of the QNF. We emphasize that this dynamical system, phase space approach to quantum reactive scattering through the QNF provides a completely new approach to the computation of the relevant quantum scattering quantities (e.g., CRP, resonances) which shows promise in leading to computationally efficient methods for "high-dimensional" systems.