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The reasons why nuclear electron-capture rates in stars depend on temperature and density are discussed, and some astrophysical applications of continuum electron-capture rates are reviewed. The modern theory of nuclear BETA decay is then used to calculate stellar-continuum electroncapture rates for transitions of an arbitrary degree of forbiddenness. The equations that are most useful for astrophysical applications are discussed in detail; particular emphasis is placed upon methods for predicting stellar rates that utilize, whenever possible, terrestrial measurements. Three examples are discussed that illustrate the use of the formulas given; the examples are: the electroncapture lifetime of a proton, the stellar BETA decay of Mo, and the effect of forbidden transitions on the abundances of elements in the iron peak. (auth)