Octav Cornea, a mathematician at the University of Montreal, presents this talk in the Joint IAS/Princeton Symplectic Geometry Seminar, held in the Simonyi Classroom at the Institute for Advanced Study. He introduces categorical approximability, a recent notion that generalizes the idea of precompactness for metric spaces, and applies it to spaces of Lagrangian submanifolds equipped with the spectral metric. Working examples include exact Lagrangians in cotangent disk bundles and equators on the 2-sphere, cases where the spaces turn out to be approximable even though they fail to be precompact in the usual sense. Cornea walks through the definitions, the motivating examples, and the gap between approximability and precompactness, drawing on joint work with Giovanni Ambrosioni and Paul Biran. The talk assumes familiarity with symplectic geometry and Lagrangian Floer theory and runs as a standard research seminar, chalk-and-slides style, aimed at specialists following current developments in the field.