Deformation theory, homological algebra and mirror symmetry

This chapter explains one of the relations between deformation theory and mirror symmetry. Deformation theory or theory of moduli is related to mirror symmetry in many ways. Moduli are related to the definitions of schemes, stacks and their complex analytic analogues, and also to various parts of homological and homotopical algebra. The chapter states precisely the conjectured coincidence of the moduli spaces by reviewing various basic aspects of moduli theory. It explains how the homological algebra can be applied to the problem of moduli and explains algebraic formalism rather than the basic geometric-analytic construction which is essential to give examples of the algebraic formalism. The chapter also includes the classical deformation theory of the holomorphic structures of vector bundles on complex manifolds.