We show that the subdiffraction imaging behavior observed in the magnifying superlens experiment, [Smolyaninov et al., Science 317, 1699 (2007)] is due to the hyperbolic dispersion of Dyakonov plasmons supported by the device. These states which are a special case of Dyakonov surface waves [Dyakonov, Sov. Phys. JETP 67, 714 (1988)] exist at the interface of a metal and an anisotropic dielectric giving rise to plasmon beams on resonance. Furthermore, we describe how the unique properties of metamaterial Dyakonov plasmons can be tailored for plasmonic devices.