  We study periodic lattices, such as vortex lattices in type II
superconductors in a random pinning potential.
  For the static case we review the prediction that the phase diagram of such
systems consists of a topologically ordered Bragg glass phase, with quasi long
range translational order, at low fields. This Bragg glass phase undergoes a
transition at higher fields into another glassy phase, with dislocations, or a
liquid. This proposition is compatible with a large number of experimental
results on BSCCO or Thalium compounds. Further experimental consequences of our
results and relevance to other systems will be discussed.
  When such vortex systems are driven by an external force, we show that, due
to periodicity in the direction transverse to motion, the effects of static
disorder persist even at large velocity. In $d=3$, at weak disorder, or large
velocity the lattice forms a topologically ordered glass state, the ``moving
Bragg glass'', an anisotropic version of the static Bragg glass. The lattice
flows through well-defined, elastically coupled, static channels. We determine
the roughness of the manifold of channels and the positional correlation
functions. The channel structure also provides a natural starting point to
study the influence of topological defects such as dislocations. In $d=2$ or at
strong disorder the channels can decouple along the direction of motion leading
to a ``smectic'' like flow. We also show that such a structure exhibits an
effective transverse critical pinning force due to barriers to transverse
motion, and discuss the experimental consequences of this effect.
