  The article covers the one-dimensional Kardar-Parisi-Zhang equation, weak
drive limit, universality, directed polymers in a random medium, replica
solutions, statistical mechanics of line ensembles, and its generalization to
several components which is used to study equilibrium time correlations of
anharmonic chains and of the discrete nonlinear Schr\"{o}dinger equation. The
notes are an extended version of my lectures at Les Houches July 2015.
