Our knowledge of non-linear genetic effects on complex traits remains limited, in part, due to the modest power to detect such effects. While kernel-based tests offer a versatile approach to test for non-linear relationships between sets of genetic variants and traits, current approaches cannot be applied to Biobank-scale datasets containing hundreds of thousands of individuals. We propose, FastKAST, a kernel-based approach that can test for non-linear effects of a set of variants on a quantitative trait. FastKAST provides calibrated hypothesis tests while enabling analysis of Biobank-scale datasets with hundreds of thousands of unrelated individuals from a homogeneous population. We apply FastKAST to 53 quantitative traits measured across ≈ 300 K unrelated white British individuals in the UK Biobank to detect sets of variants with non-linear effects at genome-wide significance.