Due to the ever-growing interest in composite materials designed with complex microstructures and capable of possessing exotic properties, it has become increasingly important to be able to accurately capture the interplay between the macroscopic response and the underlying microstructure, as the former is greatly influenced by the latter. Of the different approaches available, this thesis is concerned with the use of nonlinear homogenization to study the effective response of composites. We look to illustrate the effect that constitutive assumptions have on the methods by which such estimates can be obtained, and on the actual effective, or homogenized, response of the material. For materials whose constitutive response is governed by convex functions, we show how the convexity itself can be utilized to obtain rigorous bounds and improved estimates. We prove the optimality of variational linear comparison bounds over the class of nonlinear anisotropic composites with linearly isotropic response and introduce a new symmetric fully optimized second-order method which is able to generate estimates for the effective response of nonlinear composites. On the other hand, convexity is often inconsistent with certain physical requirements (e.g. objectivity). Such is the case of hyperelasticity, where the lack of convexity of the stored-energy functions has long been known to lead to the development of instabilities. We present a framework for studying the post-bifurcation response of such systems and apply it to a specific class of reinforced hyperelastic composites under general three-dimensional loading conditions. We also consider the class of magneto-elastic composites, which consist of hyperelastic materials that are also magnetically susceptible. Unlike in the case of hyperelasticity, there lacks a complete mathematical framework for obtaining the effective response of such materials, and researchers have only begun to investigate the potential for instabilities in these materials. We therefore generalize the same methodology used successfully in the purely mechanical context to study the post-bifurcation behavior of magneto-elastic composites. This in part requires a rigorous generalization of the theoretical aspects that underlie the method. We then calculate the post-bifurcation response of a magneto-elastic material under general plane-strain loading conditions with a magnetic field applied in the plane of deformation.