  The time-integrated current of the TASEP has non-Gaussian fluctuations of
order $t^{1/3}$. The recently discovered connection to random matrices and the
Painlev\'e II Riemann-Hilbert problem provides a technique through which we
obtain the probability distribution of the current fluctuations, in particular
their dependence on initial conditions, and the stationary two-point function.
Some open problems are explained.
