Woven and braided polymer composite structures are used in many primary aerospace applications because of their superior behavior under dynamic loading conditions and light weight. Characterization of all anisotropic properties under various strain rate and temperature conditions becomes essential for analysis, design, and numerical simulations. This paper aims to present a review of critical testing methods of polymer and composite materials. In the second part, a review of numerical and analytical models for the dynamic analysis of woven and braided composites is presented. This review article cites 138 references.