Topological Data Analysis (TDA) is a family of mathematical tools developed to robustly reveal implicit structural patterns in datasets. Among these tools, persistent homology is of particular importance. It provides a multiscale description of the topological features – such as connected components, loops or cavities – that exists in the dataset. This information is often summarized in concise descriptors such as persistence diagrams. However, computing these descriptors can often be computationally costly both in time and memory. This thesis tackles two main challenges: leveraging these descriptors in a context of data analysis, in particular to improve the preservation of topological structures during dimensionality reduction processes; and designing algorithms that efficiently compute topological descriptors leveraging specific hypothesis on the data. To this end, we first propose a new dimensionality reduction approach that builds on Topological Autoencoders, providing a new formulation that aims to favor the preservation of high-dimensional cycles. We show that they are indeed preserved – whenever possible – both quantitatively (no self-intersections) and quantitatively (preservation of the persistence diagrams for the Wasserstein distance). In addition, we introduce an algorithm specialized in the computation of the Rips persistence of planar point sets, enabling a substantial acceleration of the overall method. Secondly, we interest in the Delaunay–Rips filtration for low-dimensional Euclidean point sets. In particular, we study the instabilities it creates in the persistence diagrams with respect to perturbation of the input. We also design a dedicated algorithm that leverages the specific structure of that filtration. Finally, we investigate 2-parameter persistent homology, which is a recent research area aiming at improving the robustness to outliers. We propose an adaptation of the Flood filtration to this 2-parameter context, enabling fast computation of related descriptors. We evaluate our construction on classification tasks on synthetic and real time series datasets.