In this talk, I will first recall a notion of approximability introduced by Turing in the study of metric groups in 1938. This notion admits a natural categorification for triangulated categories whose sets of objects are endowed with a metric structure. Besides discussing briefly this notion, I will explain how this notion applies naturally to some questions in symplectic topology, in particular to the study of certain classes of Lagrangian submanifolds. It turns out that many of the familiar classes of such submanifolds - such as exact, closed Lagrangians in the cotangent bundle, equators on the 2-sphere - are approximable in this sense with interesting geometric consequences. This talk is based on work in progress with Giovanni Ambrosioni and with Paul Biran both from ETH.