[for zoom link, please email s.nagyATqmul.ac.uk] We formulate conjectures relating the geometry of conformal manifolds to the spectrum of local operators in conformal field theories in d>2 spacetime dimensions. We focus on conformal manifolds with limiting points at infinite distance with respect to the Zamolodchikov metric. Our central conjecture is that all theories at infinite distance possess an emergent higher-spin symmetry, generated by an infinite tower of currents whose anomalous dimensions vanish exponentially in the distance. Our conjectures are related to the Distance Conjecture in the swampland program. We discuss the supporting evidence, their holographic interpretation, and implications for superconformal field theories.