Post-election audits and investigations can produce more transparent, trustworthy, and secure elections. However, such investigations are limited in cases by inadequate tools and methods, an absence of meaningful evidence, and high costs. In this dissertation, I address these concerns in the following three lines of research. First, I describe my research on verifiable and transparent random sample selection for post-election audits. I investigate how counties have typically approached random sample selection, and I analyze the implications and limitations of those approaches. I propose a sampling method that has since found use in counties across the country. Second, I describe a novel approach for logging events in direct recording electronic (DRE) voting systems. My approach gives investigators more meaningful evidence about the behavior of DREs on election day. In particular, I propose to record interactions between the voter and the voting machine such that they can be replayed by investigators while preserving the anonymity of the voter. Last, I describe a novel process for efficiently verifying elections that use optical scan voting systems. My process uses image superposition to let an investigator visualize the content of many ballot images simultaneously while allowing individual treatment of anomalous ballots. I evaluate this process and demonstrate an order of magnitude improvement in the time it takes to inspect ballot images individually. This approach will let investigators more cost-effectively verify that all ballots have been accurately counted as intended by the voters.