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In this review we give a detailed introduction to the theory of (curved) $L_\infty$-algebras and $L_\infty$-morphisms. In particular, we recall the notion of (curved) Maurer-Cartan elements, their equivalence classes and the twisting procedure. The main focus is then the study of the homotopy theory of $L_\infty$-algebras and $L_\infty$-modules. In particular, one can interpret $L_\infty$-morphisms and morphisms of $L_\infty$-modules as Maurer-Cartan elements in certain $L_\infty$-algebras, and we show that twisting the morphisms with equivalent Maurer-Cartan elements yields homotopic morphisms., Comment: 62 pages, comments are welcome!

In this paper we propose a reduction scheme for polydifferential operators phrased in terms of $L_\infty$-morphisms. The desired reduction $L_\infty$-morphism has been obtained by applying an explicit version of the homotopy transfer theorem. Finally, we prove that the reduced star product induced by this reduction $L_\infty$-morphism and the reduced star product obtained via the formal Koszul complex are equivalent., Comment: 27 pages, comments are welcome!