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This work studies high-temperature superconducting spiral resonators as a viable candidate for realization of RF/microwave metamaterial atoms. The theory of superconducting spiral resonators is discussed in detail, including the mechanism of resonance, the origin of higher order modes, the analytical framework for their determination, the effects of coupling scheme, and the dependence of the resonance quality factor and insertion loss on the parity of the mode. All the aforementioned models are compared with the experimental data from a microfabricated YBa2Cu3O7-δ (YBCO) spiral resonator. Moreover, the evolution of the resonance characteristics for the fundamental mode with variation of the operating temperature and applied RF power is experimentally examined, and its implications for metamaterial applications are addressed.