  Let $X$ be a compact K\"ahler manifold. We prove that the K\"ahler-Ricci flow
starting from arbitrary closed positive $(1,1)$-currents is smooth outside some
analytic subset. This regularity result is optimal meaning that the flow has
positive Lelong numbers for short time if the initial current does. We also
prove that the flow is unique when starting from currents with zero Lelong
numbers.
