With the increasing prevalence of modern complex data in non-Euclidean (e.g., manifold) forms, there is a growing need for developing models and theory for inference of non-Euclidean data. This talk first presents some recent advances in nonparametric inference on manifolds and other non-Euclidean spaces. The initial focus is on nonparametric inference base on Fr ́echet means. In particular, we present omnibus central limit theorems for Fr ́echet means for inference, which can be applied to general metric spaces including stratified spaces, greatly expanding the current scope of inference. A robust framework based on the classical idea of median-of-means is then proposed which yields estimates with provable robustness and improved concentration. In addition to inferring i.i.d data, we also consider nonparametric regression problems where predictors or responses lying on manifolds. Various simulated or real data examples with data from various manifolds are considered.