We show that (meta)materials with hyperbolic dispersion (where two eigenvalues of the dielectric permittivity tensor have opposite signs) exhibit a broad bandwidth singularity in the photonic density of states, with resulting dramatic change in a variety of phenomena, from spontaneous emission to light propagation and scattering. Waves in such "electromagnetic hyperspace" do not suffer from the diffraction limit on the optical resolution, leading to the hyperlens - the device capable of producing magnified far-field images of subwavelength objects.