Beltrami Equation with Coefficient in Sobolev and Besov Spaces Our goal in this work is to present some function spaces on the complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f (z)$, have first derivatives locally in $X(\mathbb C)$, provided that the Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$. Keywords:quasiregular mappings, Beltrami equation, Sobolev spaces, CalderÃ³n-Zygmund operatorsCategories:30C62, 35J99, 42B20