The use of voting schemes based on rankings of alternatives to solve social choice problems can often impose significant burden on voters, both in terms of communication and cognitive requirements. In this paper, we develop techniques for preference elicitation in voting settings (i.e., vote elicitation) that can alleviate this burden by minimizing the amount of preference information needed to find (approximately or exactly) optimal outcomes. We first describe robust optimization techniques for determining winning alternatives given partial preference information (i.e., partial rankings) using the notion of minimax regret. We show that the corresponding computational problem is tractable for some important voting rules, and intractable for others. We then use the solution to the minimax-regret optimization as the basis for vote elicitation schemes that determine appropriate preference queries for voters to quickly reduce potential regret. We apply these techniques to multi-winner social choice problems as well, in which a slate of alternatives must be selected, developing both exact and greedy robust optimization procedures. Empirical results on several data sets validate the effectiveness of our techniques.