Analytical models are developed to predict the transient elastic response of fully clamped circular composite plates subject to high-velocity impact by a rigid spherical projectile. The models are based on first-order shear deformation plate theory and account for the effects of large deformations as well as propagation and reflection of flexural waves. Analytical predictions of plate deflection history and peak strain in the plates are found in good agreement with those obtained from detailed explicit FE simulations. The dynamic response is found to be governed by four non-dimensional parameters and two characteristic regimes of behaviour are identified. The models are used to construct maps to design impact-resistant composite plates.