The project advances the regularity theory of elliptic and parabolic differential and integro-differential operators with degenerate coefficients. The main emphasis is on elliptic differential operators in divergence form with non-negative definite matrix-valued weights and integro-differential operators with non-negative kernels. We aim for higher integrability and Hölder continuity of solutions resp. their derivatives in settings, where classical ellipticity assumptions fail. We complement this by investigations of possible Lavrentiev phenomenons. Building upon our regularity results, we develop and analyze numerical methods including optimal adaptive approximation schemes.