Probing, analyzing and visualizing the shape of complex data are challenges that are magnified by the intricate dependence of their structural properties on location and scale.  Thus, resolving and integrating the geometry of data across scales and over the entire data landscape are problems of primary relevance. We introduce the notion of multiscale covariance tensor fields (CTF) and show that many properties of the shape of data become more accessible through these tensor fields. Localized forms of covariance have been used empirically in data analysis, but we present a systematic treatment that includes stability theorems with respect to optimal transport metrics that ensure that properties of probability measures derived from CTFs are robust, as well as convergence results that guarantee that shape properties can be estimated reliably from data. We also discuss applications to manifold clustering, where the goal is to categorize data points according to their noisy membership in a finite union of possibly intersecting smooth manifolds.