Ballistic impact onto flexible woven textile fabrics is a complicated multi-scale problem given the structural hierarchy of the materials, anisotropic material behavior, projectile geometry-fabric interactions, impact velocity and boundary conditions. Although this subject has been an active area of research for decades, the fundamental mechanisms such as material failure, dynamic response, multi-axial loading occurring at the lower length scales during impact are not well understood. This paper reviews the recent advances in modeling and experiments of Kevlar ballistic fibrils, fibers, yarns and flexible woven textile fabrics pertinent to the deformation modes occurring during impact and serves to identify topics worthy of further investigation that will advance the basic understanding of the phenomena governing transverse impact. This review also explores on aspects such as homogeneous versus heterogeneous behavior of yarns consisting of individual fibers and the inelastic transverse behavior of the fiber which is not considered in the previous review papers on this topic.