  These lecture notes, adapted from the habilitation thesis of the author,
survey in a first part various exact results obtained in the past few decades
about KPZ fluctuations in one dimension, with a special focus on finite volume
effects describing the relaxation to its stationary state of a finite system
starting from a given initial condition. The second part is more specifically
devoted to an approach allowing to express in a simple way the statistics of
the current in the totally asymmetric simple exclusion process in terms of a
contour integral on a compact Riemann surface, whose infinite genus limit leads
to KPZ fluctuations in finite volume.
