The central limit theorem (CLT) is commonly thought of as

occurring on the real line, or in multivariate form on a

real vector space. Motivated by statistical applications

involving nonlinear data, such as angles or phylogenetic

trees, the past twenty years have seen CLTs proved for

Fréchet means on manifolds and on certain examples of

singular spaces built from flat pieces glued together in

combinatorial ways. These CLTs reduce to the linear case

by tangent space approximation or by gluing. What should a

CLT look like on general non-smooth spaces, where tangent

spaces are not linear and no combinatorial gluing or flat

pieces are available? Answering this question involves

figuring out appropriate classes of spaces and measures,

correct analogues of Gaussian random variables, how the

geometry of the space (think "curvature") is reflected in

the limiting distribution, and generally how the geometry

of sampling from measures on singular spaces behaves. This

talk provides an overview of these answers, starting with a

review of the usual linear CLT and its generalization to

smooth manifolds, viewed through a lens that casts the

singular CLT as a natural outgrowth, and concluding with

how this investigation opens gateways to further advances

in geometric probability, topology, and statistics. Joint

work with Jonathan Mattingly and Do Tran.