Local $C^{k,\beta }$ and $W^{2 + k,v}$ ($k \geq 1,\beta > 0$, and $v \geq 1$) regularity is established for the solutions of a class of degenerate quasilinear elliptic equations, which include the p-Laplacian. Unlike the known local regularity results for such equations, k is larger than 2 in many notable cases. These results generalize those in [13], which were established only for the p-Laplacian. Furthermore, local results are extended to obtain a global regularity result in some cases. Global results of this type are essential in proving optimal error bounds for the finite element approximation of such equations.