  We study current fluctuations in lattice gases in the macroscopic limit
extending the dynamic approach for density fluctuations developed in previous
articles. More precisely, we establish a large deviation principle for a
space-time fluctuation $j$ of the empirical current with a rate functional $\mc
I (j)$. We then estimate the probability of a fluctuation of the average
current over a large time interval; this probability can be obtained by solving
a variational problem for the functional $\mc I $. We discuss several possible
scenarios, interpreted as dynamical phase transitions, for this variational
problem. They actually occur in specific models. We finally discuss the time
reversal properties of $\mc I$ and derive a fluctuation relationship akin to
the Gallavotti-Cohen theorem for the entropy production.
