Engineering optical properties using artificial nanostructured media known as metamaterials has led to breakthrough devices with capabilities from super-resolution imaging to invisibility. In this paper, we review metamaterials for quantum nanophotonic applications, a recent development in the field. This seeks to address many challenges in the field of quantum optics using advances in nanophotonics and nanofabrication. We focus on the class of nanostructured media with hyperbolic dispersion that have emerged as one of the most promising metamaterials with a multitude of practical applications from subwavelength imaging, nanoscale waveguiding, biosensing to nonlinear switching. We present the various design and characterization principles of hyperbolic metamaterials and explain the most important property of such media: a broadband enhancement in the electromagnetic density of states. We review several recent experiments that have explored this phenomenon using spontaneous emission from dye molecules and quantum dots. We finally point to future applications of hyperbolic metamaterials, using the broadband enhancement in the spontaneous emission to construct single-photon sources.