In my thesis, we attack this problem from several different angles. We present several techniques to approximate the topological summary. Some of our techniques are tailored towards point clouds in Euclidean space and make use of high-dimensional lattice geometry. We present the first polynomial-sized approximation schemes which are independent of the ambient dimension. On the other hand, we also show that polynomial complexity is impossible when a high approximation quality is desired. In addition, we also develop results for metric spaces which have low intrinsic dimension.