We introduce the concept of a twisting element based on a bialgebra and show how it can be used to twist a large class of algebras, coalgebras and certain subcategories of their respective module and comodule categories. We prove that this subcategory of modules over the original algebra is equivalent to the corresponding category of modules over the twisted algebra. The relation between twisting elements and universal deformation formulas is also given, along with new formulas which are associated to enveloping algebras of non-abelian Lie algebras.

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We develop methods for computing Hochschild cohomology groups and deformations of crossed product rings. We use these methods to ÿnd deformations of a ring associated to a particular orbifold with discrete torsion, and give a presentation of the center of the resulting deformed ring. This connects with earlier calculations by Vafa and Witten of chiral numbers and deformations of a similar orbifold.

We give a selective survey of topics in algebraic deformation theory ranging from its inception to current times. Throughout, the numerous contributions of Murray Gerstenhaber are emphasized, especially the common themes of cohomology, infinitesimal methods, and explicit global deformation formulas.