  These lecture notes discuss several related features of the exactly solvable
two-dimensional corner growth model with exponentially distributed weights. A
key property of this model is the availability of a fairly explicit stationary
version that possesses useful independence properties. With the help of
couplings and estimates, we prove the existence of Busemann functions for this
model, and the precise values of the longitudinal and transversal fluctuation
exponents for the stationary corner growth model. The Busemann functions in
turn furnish extremals for variational formulas that describe limiting shape
functions.
