Advances in computing technology have spurred two extraordinary phenomena in science: large-scale and high-throughput data collection coupled with the creation and implementation of complex statistical algorithms for data analysis. These two phenomena have brought about tremendous advances in scientific discovery but have raised two serious concerns. The complexity of modern data analyses raises questions about the reproducibility of the analyses, meaning the ability of independent analysts to recreate the results claimed by the original authors using the original data and analysis techniques. Reproducibility is typically thwarted by a lack of availability of the original data and computer code. A more general concern is the replicability of scientific findings, which concerns the frequency with which scientific claims are confirmed by completely independent investigations. Although reproducibility and replicability are related, they focus on different aspects of scientific progress. In this review, we discuss the origins of reproducible research, characterize the current status of reproducibility in public health research, and connect reproducibility to current concerns about the replicability of scientific findings. Finally, we describe a path forward for improving both the reproducibility and replicability of public health research in the future.