We design mechanisms that robustly implement any desired social choice function when (i) agents must incur a cost to learn the state of the world, (ii) with small probability, agents' preferences can be arbitrarily different from some baseline known to the social planner, and (iii) the planner does not know agents' beliefs and higher-order beliefs about one another's preferences. The mechanisms we propose have a natural interpretation, and are robust to trembles in agents' reporting strategies, to the introduction of a small amount of noise affecting agents' signals about the state, and to uncertainty concerning the state distribution and agents' prior beliefs about the state. We also establish impossibility results for stronger notions of robust implementation.