When deep-water surface gravity waves traverse an area with a curved or otherwise variable current, the current can act analogously to an optical lens, to focus wave action into a caustic region. In this region, waves of surprisingly large size, alternatively called freak, rogue, or giant waves are produced. We show how this mechanism produces freak waves at random locations when ocean swell traverses an area of random current. When the current has a constant (possibly zero) mean with small random fluctuations, we show that the probability distribution for the formation of a freak wave is universal, that is, it does not depend on the statistics of the current, but only on a single distance scale parameter, provided that this parameter is finite and non-zero. Our numerical simulations show excellent agreement with the theory, even for current standard deviation as large as 1.0 m s−1. Since many of these results are derived for arbitrary dispersion relations with certain general properties, they include as a special case previously published work on caustics in geometrical optics.