Moduli theory of algebraic varieties and classification theory of compact complex spaces -- Moduli spaces for polarized algebraic varieties -- Group quotients in the category of analytic spaces and the category of algebraic spaces -- Applications of the quotient theorems to moduli of algebraic varieties -- Quotients for affine schemes by reductive algebraic groups -- Quotients in the category of schemes -- Mumfordโ{128}{153}s construction of the moduli variety for curves and polarized abelian varieties. Other applications of Mumfordโ{128}{153}s quotient theory -- Other methods of treating moduli problems. Artinโ{128}{153}s method of algebraic stacks. Griffithsโ{128}{153}s method of period maps -- Compactification of moduli spaces -- Fine moduli spaces. The universal families for stable curves with level n-structure -- Applications of moduli theory to fibre spaces and the additivity formula for the Kodaira dimension of fibre spaces. Open problems