We consider classification problems for manifolds and discrete subgroups of Lie groups from a descriptive set theoretic point of view. This work, joint with Iian Smythe, is largely foundational in conception and character, recording both a framework for general study and Borel complexity computations for some of the most fundamental classes of manifolds. Sample computations include: 2-manifolds up to homeomorphism, hyperbolic n-manifolds up to isometry, and those with finitely g enerated f undamental group up to isometry, in the process, we strengthen results of Stuck–Zimmer, Andretta–Camerlo–Hjorth, Hjorth–Kechris, and Canary–Storm. Part of the interest of this work is the abundance of good questions it opens on to, we are eager to share these as well.