Explore a comprehensive lecture on the classification of UV completions of integrable TTbar deformations of Conformal Field Theory (CFT). Delivered by Changrim Ahn from Ewha Womans University as part of the Talking Integrability: Spins, Fields, and Strings conference at the Kavli Institute for Theoretical Physics. Delve into advanced topics in mathematical physics, condensed-matter physics, and string theory, focusing on integrable models and their recent developments. Gain insights into integrable deformations of CFTs and IQFTs, integrable spin-chains and non-equilibrium dynamics, integrable structures on string worldsheets, and their connections to AdS/CFT correspondence. Benefit from a pedagogical overview suitable for integrability practitioners from various backgrounds, bridging the gap between different fields of study in this hour-long presentation.