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If K is a totally real algebraic number field then PSL(2, D K ), with D K the ring of integers of K, is called the Hubert modular group (of K). If the degree of K over ℚ is n the n embeddings K ↪ ℝ define an embedding of PSL(2, K) into PSL(2, ℝ)n. Since PSL(2, ℝ) is the holomorphic automorphism group of the complex upper half plane ℌ the Hilbert modular group PSL(2, D K ), or more generally PSL(ℒ) with ℒ ⊂ K2 a projective D K -module of rank 2, acts on ℌ n and we can form the quotient. This is quasi-projective algebraic variety with finitely many isolated singularities and it is called a Hilbert modular variety. For n = 1 (K = ℚ) one finds the classical modular group and the j-line PSL(2, ℤ)\ ℌ. This book deals with the case where the field K is a real quadratic field.