A major challenge in the field of correlated electrons is the computation of dynamical correlation functions. For comparisons with experiments, one is interested in their real-frequency dependence. This is difficult to compute because imaginary-frequency data from the Matsubara formalism require analytic continuation, a numerically ill-posed problem. In this talk, I will show how to apply quantum field theory to the single-impurity Anderson model using the Keldysh instead of the Matsubara formalism with direct access to the self-energy and dynamical susceptibilities on the real-frequency axis. I will present results from the functional renormalization group (fRG) at the one-loop level and from solving the self-consistent parquet equations in the parquet approximation. For the first time using Keldysh fRG, we employ a parametrization of the four-point vertex, which captures its full dependence on three real-frequency arguments. We compare our results to benchmark data obtained with the numerical renormalization group and to second-order perturbation theory. We find that capturing the full frequency dependence of the four-point vertex significantly improves the fRG results compared with previous implementations and that solving the parquet equations yields the best agreement with the numerical renormalization group benchmark data but is only feasible up to moderate interaction strengths. In the final part of the talk, I will sketch how we plan to build on those results by outlining a combination of the non-perturbative but local dynamical mean-field theory (DMFT) with the diagrammatic methods from before to compute dynamical and non-local correlation functions. This will require a significant compression of the four-point vertex, and I will introduce the most promising candidate technique for that purpose, the quantics tensor cross interpolation.