In this talk,  with a broad goal of building a mathematical foundation for statistical analysis on stratified spaces, I will review definition of stratified spaces and example of data with stratified structure. We will then explore relationships between geometry and different forms of CLT, namely classical, smeary and sticky.  In particular, on Riemannian manifolds, the space’s sectional curvature can change both the rate of convergence and the form of the limit distribution of Fréchet means. On Riemannian stratified spaces, we will see that the general form of the CLT contains information about curvature and singularity of the base space.