Over the last few decades data represented in various non-conventional forms have become increasingly more prevalent. Typical examples include diffusion matrices in diffusion tensor imaging (DTI) of neuroimaging, and various digital image data arising in biology, medicine, and machine vision. One may also encounter data that are stored in the forms of subspaces, orthonormal frames, surfaces, and networks. Statistical analysis of such data requires rigorous formulation and characterization of the underlying space, and inference is heavily dependent on the geometry of the space. For a majority of the cases considered, the underlying spaces where these general data objects lie on, fall into the general category of manifolds. This talk focuses on nonparametric inference on manifolds and other non-Euclidean spaces. Appropriate notion of means and variations are defined, and inference is based on the asymptotic distributions of their sample counterparts. In particular, we present omnibus central limit theorems for Fr\'echet means for inference, from which many of the existing CLT follows immediately. Applications are provided to some stratified spaces of recent interest, and to the space of symmetric positive definite matrices arising in DTI. In addition to inferring i.i.d data, we also consider nonparametric regression problems where predictors or responses lie on manifolds.