Sparse regression has become an indispensable method for data analysis in the last 20 years. The general framework for sparse regression has a number of drawbacks that we and others address in recent methods, including robustness of model selection, issues with correlated predictors, and a test statistic that is based on the size of the effect. All of these issues arise in the context of association mapping of genetic variants to quantitative traits. This talk will discuss one approach to structured sparse regression to mitigate these problems in the context of genome-wide association mapping with quantitative traits using a Gaussian process prior to add structure to the sparsity-inducing prior across predictors. We will also describe ongoing efforts for variants on this model for different analytic purposes, including neuroscience applications, identifying driver somatic mutations in cancer, and methods for causal inference in observational data with large numbers of instruments.