  Domain walls in magnets, vortex lattices in superconductors, contact lines at
depinning, and many other systems can be modelled as an elastic system subject
to quenched disorder. Its field theory possesses a well-controlled perturbative
expansion around its upper critical dimension. Contrary to standard field
theory, the renormalization group flow involves a function, the disorder
correlator $\Delta(w)$, therefore termed the functional renormalization group
(FRG). $\Delta(w)$ is a physical observable, the auto-correlation function of
the centre of mass of the elastic manifold. In this review, we give a
pedagogical introduction into its phenomenology and techniques. This allows us
to treat both equilibrium (statics), and depinning (dynamics). Building on
these techniques, avalanche observables are accessible: distributions of size,
duration, and velocity, as well as the spatial and temporal shape. Various
equivalences between disordered elastic manifolds, and sandpile models exist:
an elastic string driven at a point and the Oslo model; disordered elastic
manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or
loop-erased random walks. Each of these mappings requires specific techniques,
which we develop, including modelling of discrete stochastic systems via
coarse-grained stochastic equations of motion, super-symmetry techniques, and
cellular automata. Stronger than quadratic nearest-neighbour interactions lead
to directed percolation, and non-linear surface growth with additional KPZ
terms. On the other hand, KPZ without disorder can be mapped back to disordered
elastic manifolds, either on the directed polymer for its steady state, or a
single particle for its decay. Other topics covered are the relation between
functional RG and replica symmetry breaking, and random field magnets. Emphasis
is given to numerical and experimental tests of the theory.
