First-passage percolation is a canonical example of a random metric on the lattice $\mathbb{Z}^d$. It is also conjecturally in the KPZ universality class for growth models. This is a three-part talk, in which we will cover the following topics:

1. Overview of geodesics in first-passage percolation; their asymptotic geometry and KPZ behavior; bigeodesics and their connections to the random Ising model.

2. Busemann functions, their construction and their properties; encoding geodesic behavior using Busemann functions.

3. Geodesic behavior from an abstract, ergodic theoretic viewpoint; geodesics as the flow lines of a random vector field.