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Summary form only given. Fractional derivatives/integrals are mathematical operators involving differentiation/integration to arbitrary noninteger orders-orders that may be fractional or even complex. These operators, which possess interesting mathematical properties, have been studied in the field of fractional calculus. In our study, we have applied the tools of fractional calculus in various problems in electromagnetic fields and waves, and have obtained interesting results that highlight certain notable features and promising potential applications of these operators in electromagnetic theory. Furthermore, since fractional derivatives/integrals are effectively the result of fractionalization of differentiation and integration operators, we have investigated the notion of fractionalization of some other linear operators in electromagnetic theory. Searching for such operator fractionalization has led us to novel solutions, interpretable as "fractional solutions", for certain electromagnetic problems. A brief review of general principles, definitions, and some of the features of fractional derivatives/integrals are given. Then we present an overview of some fractional mathematical operators involving our ideas and findings in developing the differentiation/integration to arbitrary fractional paradigm in electromagnetism and its potential applications, and we discuss some specific cases in detail. Physical insights into these results are also provided and future directions in this area are addressed.