We provide a framework for proving convergence to the directed landscape, the
central object in the Kardar-Parisi-Zhang universality class. For last passage
models, we show that compact convergence to the Airy line ensemble implies
convergence to the Airy sheet. In i.i.d. environments, we show that Airy sheet
convergence implies convergence of distances and geodesics to their
counterparts in the directed landscape. Our results imply convergence of
classical last passage models and interacting particle systems. Our framework
is built on the notion of a directed metric, a generalization of metrics which
behaves better under limits. As a consequence of our results, we present a
solution to an old problem: the scaled longest increasing subsequence in a
uniform permutation converges to the directed geodesic.

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