  We analyze the averaged controllability properties of random evolution
Partial Differential Equations.
  We mainly consider heat and Schr\"odinger equations with random parameters,
although the problem is also formulated in an abstract frame.
  We show that the averages of parabolic equations lead to parabolic-like
dynamics that enjoy the null-controllability properties of solutions of heat
equations in an arbitrarily short time and from arbitrary measurable sets of
positive measure.
  In the case of Schr\"odinger equations we show that, depending on the
probability density governing the random parameter, the average may behave
either as a conservative or a parabolic-like evolution, leading to
controllability properties, in average, of very different kind.
