Investigations of the response of composites to time-dependent loadings are of considerable importance in structural analysis of many engineering fields, including aerospace, aeronautic, naval and automotive applications. Wave propagation in woven fabric composites poses major theoretical and experimental challenges. The mathematical modeling of dynamic stress concentrations in woven fabric composites is a fairly difficult task due to the great number of wave interactions between the fiber and the matrix. When composite structures composed of woven fabric preforms are loaded impulsively, stress waves propagate in different constituent phases at different speeds, which results in dynamic normal and shearing stresses at the interfacial bond region as well as at the yarn cross-over points. The concentration of these stresses may initiate unstable delamination and eventually failure of the laminated structure. This paper presents a theoretical study on the elastic wave propagation and local dynamic stress concentration in woven fabric composites. The analysis focuses on the unit cell of an orthogonal woven fabric composite, which is composed of two sets of mutually orthogonal yarns of either the same fiber (non-hybrid fabric) or different fibers (hybrid fabric) in a matrix material. Using the mosaic model for simplifying woven fabric composites and a shear lag approach to take into account the inter-yarn deformation, a one-dimensional analysis has been developed to predict the local elastodynamic and elastostatic behavior. The initial and boundary value problems are formulated and then solved using Laplace transforms. Closed form solutions of the dynamic displacements and stresses in each yarn, and the bond shearing stresses at the interfaces between adjacent yarns, are obtained in the time domain for in-plane, as well as out-of-plane impact loadings. When time tends to infinity, the dynamic solutions approach to their corresponding static solutions, which are also developed in this article. Solutions of certain special cases are identical to those reported in the literature. Lastly, the dynamic stresses and bond shearing stresses of plain weave composites subjected to step uniform impacts are presented and discussed as an example of the general analytical model.