First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation between deformations and solutions of the corresponding Maurer-Cartan equation. In Section  we generalize the Maurer-Cartan equation to strongly homotopy Lie algebras and prove the homotopy invariance of the moduli space of solutions of this equation. In the last...

An exposition is given of recent work of the author and others on the differential calculi that occur in the setting of compact quantum groups. The principal topics covered are twisted graded traces, an extension of Connes' cyclic cohomology, invariant linear functionals on covariant calculi and the Hodge, Dirac and Laplace operators in this setting. Some new results extending the classical de Rham theorem and Poincaré duality are also discussed.