Recent years have seen a wave of laws making it more difficult to vote in the United States. Their ostensible purpose is to prevent voting by persons not legally qualified, and thus to improve public confidence in electoral integrity. Are such laws in fact needed to address serious fraud problems? On the other hand, how many and what kinds of legitimate voters will be, or have been, disenfranchised by them? Is voter confidence positively, or negatively, affected by voter ID laws? This article surveys what is known about these issues and offers suggestions for how statisticians can contribute.