In this talk, we discuss three problems in non-Euclidean data analysis. The two first problems are concerned with dimensionality reduction, on one hand for shapes in the sense of Kendall, and on the other hand for network-valued data. Such objects share in common that they are modeled as elements of a quotient space which can be equipped with a Riemannian structure. Most dimensionality reduction methods for Riemannian data consist of generalizations of PCA based on the two notions of geodesic component and geodesic subspace. We propose to go beyond this approach in two manners. To escape the rigidity of geodesic subspaces, we generalize Locally Linear Embedding (LLE), a manifold learning method, to the case of Riemannian data and in particular that of shapes. Then, we describe a new geometric model for network-valued data that is compatible with Barycentric Subspace Analysis (BSA), a better interpretable variant of PCA. In the last problem, we come back to statistical shape analysis, specifically to the analysis of n-dimensional curves. The problem is one in single-cell data analysis known as trajectory inference, namely that of reconstructing a differentiation tree from sequencing data. To this end, we show the property of varifold distances to characterize the topology of a vector field — in our case a RNA velocity field — based on its integral curves.