In this talk, I will explore the interesting geometries that emerge in high-dimensional attraction basins, which are important in applications such as protein folding, cell differentiation, and neural networks. As a paradigmatic model, I will consider networks of coupled Kuramoto oscillators and show that high-dimensional basins generally cannot be approximated by simple convex shapes. Instead, they have tentacle-like structures where most of the basin volume is concentrated. Next, I will show that introducing non-pairwise couplings among Kuramoto oscillators can make basins deeper but smaller—the attractors become linearly more stable but much harder to find due to basins shrinking dramatically. Time allowing, I will also briefly mention a few related projects, including learning basins with reservoir computing and modeling circadian clocks with Kuramoto oscillators.