This chapter discusses some of the challenges associated with the analysis of composite structures. It begins with a review of lamina fundamentals and the stress-strain relationships in a single ply under various types of loads. It demonstrates the use of classical lamination theory, discusses the effects of interlaminar free-edge stresses, and explains how to predict the failure of composites using stress and strain criteria as well as the Azzi-Tsai-Hill maximum work theory and the Tsai-Wu failure criterion.