A class of d-level quantum states called "magic states", whose initial purpose was to enable universal fault-tolerant computation within error-correcting codes, has a surprisingly broad range of applications. We begin by describing their structure with respect to the Clifford hierarchy, and in terms of convex geometry before proceeding to their applications. They appear to have some relevance to the search for SIC-POVMs in certain prime dimensions. A version of the CHSH non-local game, using a d-ary alphabet and Pauli measurements, has an optimal quantum strategy using magic states. Finally, magic states exhibit nice symmetries (balanced, minimum uncertainty) with respect to Pauli measurements and consequently could find applications in areas like cryptography.