The theory of anisotropic sputtering proposed by Zhang [Z.L. Zhang, Phys. Rev. B 71 026101 (2005).] and [Z.L. Zhang and L. Zhang, Radiat. Eff. Defects Solids 159(5) 301 (2004).] has been generalized to sputtering of isotopic mixtures. The present theory (modified Sigmund theory) has been shown to fit numerous simulations and experimental measurements, including energy and angular distribution of sputtered atoms. In particular, the theory has successfully solved the isotope puzzle of sputtering induced by low energy and heavy ion bombardment.