Many of the mathematical work on metamaterial has focused on negative index materials, with negative and isotropic dielectric coefficients. Such media can be realized as homogenized limits of high contrast composite (typically metals and di-electric media). However, homogenization limits are often anisotropic, and may yield hyperbolic metamaterials, i.e., media with anisotropic di-electric tensors with both positive and negative eigenvalues. We give some examples of constructions that use hyperbolic metamaterials, which in the limit of small absorption lead to coupled elliptic/hyperbolic PDEs, and which exhibit interesting superlensing properties. This is joint work with Hoai-Minh Nguyen (EPFL).