In lectures given in 1953 at New York University, Franz Rellich proved that for all f C 0∞(R n \{0}) and n ≠ 2 [InlineMediaObject not available: see fulltext.] where the constant C(n):=n 2(n-4)2/16 is sharp. For n=2 extra conditions were required for f, and for n=4, C(4)=0, producing a trivial inequality. Influenced by recent work of Laptev-Weidl on Hardy-type inequalities in R 2, the authors show that for n≥2, the inclusion of a magnetic field B=curl(A) of Aharonov-Bohm type yields non-trivial Rellich-type inequalities of the form [InlineMediaObject not available: see fulltext.] where Δ A =(∇-i A)2 is the magnetic Laplacian. As in the Laptev-Weidl inequality, the constant C(n,α) depends upon the distance of the magnetic flux [InlineMediaObject not available: see fulltext.] to the integers Z. When the flux [InlineMediaObject not available: see fulltext.] is an integer and α=0, the inequalities reduce to Rellich's inequality. © Springer-Verlag 2005.