In this project, we aim for a full control of quark-mass effects in Higgs production through gluon fusion at higher orders in perturbation theory. This includes the calculation of the top- and bottom-quark mass effects for the total and differential cross section at 4th order in the strong coupling and their implementation in Monte-Carlo programs. In parallel to the exact evaluation of these quantities, we will also approach the problem through approximations by evaluating and combining amplitudes in characteristic kinematical limits. The developed methods will also allow us to consider effects from potential physics beyond the Standard Model.