Understanding limits on the superconducting transition temperature Tc is a question of fundamental and practical importance. I will begin by describing quantum materials and ultracold atoms experiments that challenge conventional wisdom on what controls Tc. I will present exact upper bounds  on the superfluid stiffness, which in turn lead to bounds on the BKT Tc in 2D independent of pairing strength or mechanism. The bound takes a particularly simple form for parabolic dispersion in 2D – Tc cannot exceed one-eighth the Fermi temperature – which has been realized in recent experiments . I will next describe results for multi-band systems with complex band structures and discuss applications to 2D superconducting materials including monolayer FeSe/STO and twisted bilayer graphene. I will then describe progress on topological flat bands, where the Tc bound  involves the quantum geometry of electronic wave functions. I will conclude by describing the challenges in deriving bounds on Tc in 3D, which remains an open problem.
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