Advances in experimental technologies have unlocked a set of previously unobservable phenomena ranging from turbulent flows in environmental sciences to the coordinated motion of thousands of cells during embryonic development. These systems, defined by finite-time datasets, exhibit nonlinear, multi-scale and chaotic behaviors, and disentangling their complex paths and unveiling their mechanistic basis require new techniques and ideas. Using concepts from calculus of variations, differential geometry and Hamiltonian systems, I discuss the development of mathematical tools that unveil the intrinsic geometric organizers (or Coherent Structures) of the dynamical systems’ phase space. I illustrate these results on challenging experiments related to atmospheric flows, highly unsteady separated flows, search and rescue operations at sea, and morphogenetic multicellular flows during embryonic development. I show how these techniques uncover previously unknown structures, which include the onset of aerodynamic separation, hidden short-term attractors on the ocean surface and new biological features that shed light on when cells became fated during embryogenesis. I will conclude with ongoing and future research directions.