  We present a review of our numerical studies on non equilibrium vortex
dynamics in Josephson Junction arrays (JJA) driven by a dc current. Dynamical
phase diagrams for different magnetic fields, current directions and varying
temperature are discussed and compared.
  First, the effect of thermal fluctuations in a current driven diluted vortex
lattice (VL) with f=1/25 is analyzed. The more interesting studied regime is
the one for large currents where there is sequence of transitions for the
transverse response of a fast moving VL. A transverse depinning temperature
below the melting transition of the moving VL is obtained.
  We also discuss the dependence with the direction of the applied dc current
of the transport properties of diluted VL at low temperatures. We show that
orientational pinning phenomenon leads to a finite transverse critical current
when the bias current is applied in the directions of high symmetry and it
leads to an anomalous transverse voltage when vortices are driven away from the
favorable directions. In addition, the effect of disorder with a diagonal dc
current applied is analyzed and a finite transverse voltage is also observed in
this case.
  The fully frustrated JJA (f=1/2) case is also discussed. In equilibrium, the
low temperature phase has two broken symmetries: the U(1) symmetry and the Z_2
symmetry. At high currents two well separated transitions are observed. The
order of the checkerboard VL (discrete Z_2 symmetry) is destroyed at a much
lower temperature than the transverse superconducting coherence (continuous
U(1) symmetry).
