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Confining electromagnetic energy is crucial to enhance light-matter interactions, with important implications for science and technology. Here, the opportunities offered by trapping and confining light in open structures, based on the concept of embedded eigenstates within the radiation continuum enabled by zero-index metamaterials, are discussed. Building upon the physical insights offered by the analysis, a general platform is put forward that allows the realization of extremely high field enhancements in open structures under external illumination. Structures supporting embedded eigenstates represent a rare example of physical systems in which extreme-in principle unbounded-responses can be tamed. The proposed design recipe to realize bound states in the continuum also offers a simple model that allows testing of important questions that surround the concept of embedded eigenstates, such as their effect on the local density of photonic states. The findings help clarify which nano-optical and radio-wave applications may benefit from this unusual and singular response. [ABSTRACT FROM AUTHOR]