The stationary horizon (SH) is a stochastic process of coupled Brownian motions indexed by their real-valued drifts. It was first introduced Busani as the diffusive scaling limit of the Busemann process of exponential last-passage percolation. It was independently discovered as the Busemann process of Brownian last-passage percolation by the Sepp\"al\"ainen and Sorensen. We show that SH is the

unique invariant distribution and an attractor of the KPZ fixed point under conditions on the asymptotic spatial slopes. It follows that SH describes the Busemann process of the directed landscape. This gives control of semi-infinite

geodesics simultaneously across all initial points and directions. The countable dense set $\Xi$ of directions of discontinuity of the Busemann process is the set of directions in which not all geodesics coalesce and in which there exist at least two distinct geodesics from each initial point. This creates two distinct families of coalescing geodesics in each $\Xi$ direction. In $\Xi$ directions, the Busemann difference profile is distributed as Brownian local time. We describe the point process of directions $\xi\in\Xi$ and spatial locations where the $\xi\pm$ Busemann functions separate. Based on joint work with Ofer Busani and Timo Sepp\"al\"ainen.