We give a survey on generalized Krein algebras <$E K sub p,q sup {alpha , beta}> and their applications to Toeplitz determinants. Our methods originated in a paper by Mark Krein of 1966, where he showed that <$E K sub 2,2 sup {1 "/" 2 , 1 "/" 2}> is a Banach algebra. Subsequently, Widom proved the strong Szego limit theorem for block Toeplitz determinants with symbols in <$E (K sub 2,2 sup {1 "/" 2 , 1 "/" 2}) sub {N times N}> and later two of the authors studied symbols in the generalized Krein algebras <$E (K sub p,g sup {alpha , beta}) sub {N times N}>, where <$E lambda~:=~1 "/" p~+~1 "/" q~=~alpha~+~beta> and <$E lambda~=~1>. We here extend these results to <$E 0~<<~lambda~<<~1>. The entire paper is based on fundamental work by Mark Krein, ranging from operator ideals through Toeplitz operators up to Wiener - Hopf factorization.

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