  Talk presented at the International Conference on Mathematical Physics
(Brisbane 1997). This is an introduction to recent work on the scaling and
intermittency in forced Burgers turbulence. The mapping between Burgers'
equation and the problem of a directed polymer in a random medium is used in
order to study the fully developped turbulence in the limit of large
dimensions. The stirring force corresponds to a quenched (spatio temporal)
random potential for the polymer, correlated on large distances. A replica
symmetry breaking solution of the polymer problem provides the full probability
distribution of the velocity difference $u(r)$ between points separated by a
distance $r$ much smaller than the correlation length of the forcing. This
exhibits a very strong intermittency which is related to regions of shock
waves, in the fluid, and to the existence of metastable states in the directed
polymer problem. We also mention some recent computations on the finite
dimensional problem, based on various analytical approaches (instantons,
operator product expansion, mapping to directed polymers), as well as a
conjecture on the relevance of Burgers equation (with the length scale playing
the role of time) for the description of the functional renormalisation group
flow for the effective pinning potential of a manifold pinned by impurities.
