Experimental measurements of biological systems often have a limited number of independent channels, hindering the construction of interpretable models of living processes. Dynamical systems theory provides a rich set of tools for inferring underlying mathematical structure from partial observations, yet translating these insights to real-world biological datasets remains challenging. In this talk, I will overview my recent work at the intersection of nonlinear dynamics, chaos, and biology. I will first focus on my recent work on developing physics-informed machine learning algorithms that extract dynamical models directly from raw experimental data. I will present a general technique for inferring strange attractors directly from diverse biological time series, including gene expression, patient electrocardiograms, fitness trackers, and neural spiking. Next, I will next discuss my efforts to apply concepts from dynamical systems theory to understand particular biological phenomena. I will describe my work on biological fluid dynamics, and the discovery of a beautiful vortex crystal formed by the swimming strokes of early-diverging animals—which enables a novel feeding strategy based on chaotic mixing of the local microenvironment. I will relate this work to broader questions at the intersection of nonlinear dynamics and organismal behavior. I will conclude by discussing how these insights open up several exciting new avenues at the intersection of dynamical systems theory, systems biology, and machine learning.