  We compare several recent approaches to studying right Bousfield localization
and algebras over monads. We prove these approaches are equivalent, and we
apply this equivalence to obtain several new results regarding right Bousfield
localizations (some classical, some new) for spectra, spaces, equivariant
spaces, chain complexes, simplicial abelian groups, and the stable module
category. En route, we provide conditions so that right Bousfield localization
lifts to categories of algebras, so that right Bousfield localization preserves
algebras over monads, and so that right Bousfield localization forms a
compactly generated model category.
