Composite structures, such as laminated beams, plates and shells, are widely used in the automotive, aerospace and marine industries due to their superior specific strength and tailor-able mechanical properties. Because of their use in a wide range of applications, and their commonplace in the engineering design community, the need to accurately predict their behavior to external stimuli is crucial. We consider in this thesis the application of the least-squares finite element method (LSFEM) to problems of static deformations of laminated and sandwich plates and transient plane stress deformations of sandwich beams. Models are derived to express the governing equations of linear elasticity in terms of layer-wise continuous variables for composite plates and beams, which allow inter-laminar continuity conditions at layer interfaces to be satisfied. When Legendre-Gauss-Lobatto (LGL) basis functions with the LGL nodes taken as integration points are used to approximate the unknown field variables, the methodology yields a system of discrete equations with a symmetric positive definite coefficient matrix. The main goal of this research is to determine the efficacy of the LSFEM in accurately predicting stresses in laminated composites when subjected to both quasi-static and transient surface tractions. Convergence of the numerical algorithms with respect to the LGL basis functions in space and time (when applicable) is also considered and explored. In the transient analysis of sandwich beams, we study the sensitivity of the first failure load to the beam's aspect ratio (AR), facesheet-core thickness ratio (FCTR) and facesheet-core stiffness ratio (FCSR). We then explore how failure of sandwich beams is affected by considering facesheet and core materials with different in-plane and transverse stiffness ratios. Computed results are compared to available analytical solutions, published results and those found by using the commercial FE software ABAQUS where appropriate