Computer experiments provide a convenient framework for investigating real world phenomena, and making more information available to a simulation enables it to represent reality more accurately. However, many complex simulations suffer from the curse of dimensionality, which makes an accurate simulation difficult and requires the use of a model surrogate. In many cases, a simulation mainly depends only on a subset of the input parameters and is therefore a possible target for dimensionality reduction techniques. These techniques aim to identify such input parameter structures and try to adapt the simulation accordingly, for example by constructing a lower-dimensional model surrogate. Done well, this approach can ease the curse of dimensionality while still preserving a required level of simulation accuracy. Various approaches to dimensionality reduction already exist in the context of sparse grids, such as adaptive sparse grids or sparse grid based analysis of variance, where the goal is to neglect or eliminate less important dimensions of a model function. Active subspaces are another dimensionality reduction technique, which in contrast to sparse grid based methods, can replace input parameters through linear combiniations of others, thus making them more flexible than purely dimension oriented approaches. In this thesis, the active subspace method is applied in combination with sparse grids to obtain the name-giving transformed sparse grids and create a more flexible dimensionality reduction technique that can exploit the best of both worlds. This combined approach views the model from a black-box perspective and can therefore be employed on a wide variety of high-dimensional models. To evaluate the quality and properties of the produced transformed sparse grid surrogates in different contexts, we also perform a range of experiments on a variety of high-dimensional models that are used for real world applications.