Quantum impurity problems provide fundamental descriptions of important physical problems such as quantum dots and magnetic impurities in metals, and also play an essential role in dynamical mean-field theory as the effective many-body problem that must be solved. In order to study realistic systems, features such as a generalized parametrization and multiple bands must be included. Efficient numerical quantum impurity solvers are required to treat such problems. In this project, we will develop, optimize, and apply a very promising set of methods, namely matrix-product-state-based quantum impurity solvers, which encompass the density-matrix renormalization group (DMRG) as well as the numerical renormalization group. Recent advances in these methods, especially with regard to the calculation of dynamical, i.e., frequency-dependent response functions, have the potential to significantly improve the effi- ciency and accuracy. Here we will address two topics: (i) the adaptation, development, and optimization of adaptive Chebyshev and Lanczos-vector methods and the correction vector method for the calculation of dynamical response functions for impurity problems within a matrix-product-state-based formulation of the DMRG, and (ii) the extension of the methods to efficiently treat impurities coupled to multiple bands as well as systems containing multiple impurities.