This thesis addresses the reduced basis methods for parametrized quasilinear elliptic and parabolic partial differential equations with strongly monotone differential operator. It presents all of the ingredients of the reduced basis method: basis generation for reduced basis approximation, certification of the approximation error by suitable a-posteriori error control and an Offine-Online decomposition. The methodology is further applied to the magnetostatic and magnetoquasistatic approximations of Maxwell’s equations and its validity is confirmed by numerical examples.