  In this survey article we consider the directed last-passage percolation
model on the planar square lattice with nearest-neighbor steps and general
i.i.d. weights on the vertices, outside of the class of exactly solvable
models. We show how stationary cocycles are constructed from queueing fixed
points and how these cocycles characterize the limit shape, yield existence of
Busemann functions in directions where the shape has some regularity, describe
the direction of the competition interface, and answer questions on existence,
uniqueness, and coalescence of directional semi-infinite geodesics, and on
nonexistence of doubly infinite geodesics.
