  Given a combinatorial (semi-)model category $M$ and a set of morphisms $C$,
we establish the existence of a semi-model category $L_C M$ satisfying the
universal property of the left Bousfield localization in the category of
semi-model categories. Our main tool is a semi-model categorical version of a
result of Jeff Smith, that appears to be of independent interest. Our main
result allows for the localization of model categories that fail to be left
proper. We give numerous examples and applications, related to the Baez-Dolan
stabilization hypothesis, localizations of algebras over operads, chromatic
homotopy theory, parameterized spectra, $C^*$-algebras, enriched categories,
dg-categories, functor calculus, and Voevodsky's work on radditive functors.
