  This review article discusses limit distributions and variance bounds for
particle current in several dynamical stochastic systems of particles on the
one-dimensional integer lattice: independent particles, independent particles
in a random environment, the random average process, the asymmetric simple
exclusion process, and a class of totally asymmetric zero range processes. The
first three models possess linear macroscopic flux functions and lie in the
Edwards-Wilkinson universality class with scaling exponent 1/4 for current
fluctuations. For these we prove Gaussian limits for the current process. The
latter two systems belong to the Kardar-Parisi-Zhang class. For these we prove
the scaling exponent 1/3 in the form of upper and lower variance bounds.
