This thesis is concerned with covariance operators in statistics, both as object of statistical inference (Chapters 2,3 ) and as tool for statistical inference (Chapter 4).

The first part of this thesis considers the non-linear and infinite-dimensional space of covariance operators on some abstract Hilbert space, endowed with the Bures-Wasserstein metric. Chapter 2 studies existence, uniqueness, regularity and large sample theory of Bures-Wasserstein Barycenters, in the form of a Law of Large Numbers, Central Limit Theorem and Large Deviations Principle. Then, Chapter 3 considers methodology and inference for populations of time-varying covariance operators (covariance flows) in the Bures-Wasserstein space; here, the geometric structure given by optimal transport (i.e.\ the Otto calculus) are made use of to define a notion of mean and covariance of a random flow, and develop an associated principal component analysis and Karhunen-Lo`eve expansion.

The second part of this thesis considers inference for diffusion processes, and investigates the link between the local behaviour of the processes (encapsulated in the corresponding drift and diffusion parameters) and the global behaviour of the processes (i.e., the corresponding mean and covariance kernel). Such duality is employed to establish a nonparametric inference procedure on the basis of longitudinal, sparse and noise-corrupted observations of replicated diffusion processes.