In the colored asymmetric simple exclusion process (ASEP), one places a particle of “color” $-k$ at each integer site $k \in \mathbb{Z}$. Particles attempt to swap places to the left with rate $q \in [0,1)$ and to the right with rate 1; the swap succeeds if the initiating particle has a higher color than the other particle (thus the particles tend to get more ordered over time). We will discuss the space-time scaling limit of this process (as well as a related discrete analog known as the stochastic six-vertex model), captured via a height function given by certain colored particle counts. The limit lies in the Kardar-Parisi-Zhang universality class, and is given by the Airy sheet and directed landscape, which were first constructed in 2018 by Dauvergne-Ortmann-Virág as limits in a very different setting – of fluctuations of a model of a random directed metric. The Yang-Baxter equation and line ensembles (collections of random non-intersecting curves) with certain Gibbs or spatial Markov properties will play fundamental roles in our discussion. This is based on joint work with Amol Aggarwal and Ivan Corwin.