  The Kardar-Parisi-Zhang (KPZ) universality class describes a large class of
2-dimensional models of random growth, which exhibit universal scaling
exponents and limiting statistics. The last ten years has seen remarkable
progress in this area, with the formal construction of two interrelated
limiting objects, now termed the KPZ fixed point and the directed landscape
(DL). This dissertation focuses on a third central object, termed the
stationary horizon (SH). The SH was first introduced (and named) by Busani as
the scaling limit of the Busemann process in exponential last-passage
percolation. Shortly after, in the author's joint work with Sepp\"al\"ainen, it
was independently constructed in the context of Brownian last-passage
percolation. In this dissertation, we give an alternate construction of the SH,
directly from the description of its finite-dimensional distributions and
without reference to Busemann functions. From this description, we give several
exact distributional formulas for the SH. Next, we show the significance of the
SH as a key object in the KPZ universality class by showing that the SH is the
unique coupled invariant distribution for the DL. A major consequence of this
result is that the SH describes the Busemann process for the DL. From this
connection, we give a detailed description of the collection of semi-infinite
geodesics in the DL, from all initial points and in all directions. As a
further evidence of the universality of the SH, we show that it appears as the
scaling limit of the multi-species invariant measures for the totally
asymmetric simple exclusion process (TASEP). This dissertation is adapted from
two joint works with Sepp\"al\"ainen and two joint works with Busani and
Sepp\"al\"ainen.
