The quality of genetics-based personalized medicine is a direct function of the success of statistical genomics, defined here as the application of statistical methodologies to genome data. The following dissertation provides two new statistical tools and insights for three areas of interest within the statistical genomics field: (1) better disease outcome prediction using personal genomes, (2) describing the association between genome regions and an outcome, and (3) discovering previously unknown subpopulations within a population. With respect to each of the three problems, penalized regression, in particular regression utilizing the truncated L1-penalty (TLP), is an essential element of the related methodology. Collectively, the dissertation reveals potential gains from using penalties better aligned with the data's structure and the research aim; for example, by syncing penalty features to underlying genetic architectures to improve prediction. Supported by both simulation and real data analysis, the work herein develops and demonstrates the promise of (1) a new global testing statistic for quantifying the association of a targeted genome region and a disease outcome and (2) a new group truncated L1-penalty (gTLP) methodology akin to hierarchical clustering that in some settings is able to uncover previously unknown subpopulations.