This thesis discusses the regularity of a distribution solution to nonlinear elliptic equations when the right-hand side is a measure. First, we establish Calder`{o}n-Zygmund type estimates for the borderline double phase problems by proving that the gradient of a solution has equivalent integrability to the 1-fractional maximal function of the given measure. Second, we obtained the maximal differentiability of the gradient of a solution to non-linear elliptic measure data problems with general growth.