In Euclidean space, Principal Component Analysis is widely used both for finding low dimensional approximations of sampled data and for visualization. Generalizing the procedure to non-linear manifolds raises fundamental questions about which properties of the Euclidean procedure to preserve. Based on the Eells-Elworthy-Malliavin construction of Brownian motion, I will present an approach that uses diffusion processes with anisotropic generators to construct a framework for likelihood estimation of template and covariance structure from manifold valued data. The Diffusion PCA procedure avoids the linearisation that arises when first estimating a mean or template before performing PCA in the tangent space of the mean. The anisotropic diffusion processes imply that the most probable ways of reaching sampled data will be with a different set of paths than geodesics, and we will see examples of these most probable paths.