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Humans interact with chaotic systems on everyday basis. Chaos plays a fundamental role on a wide span of length scales. It can be seen in the motion of asteroids, the formation of weather patterns, population growth, and even in the firing of neurons. Since it is hard to isolate a chaotic system from random interactions with the environment, the challenges in studying its behavior are both mathematical and experimental. In recent years, atomic gasses have emerged as experimentally accessible systems for observing chaos under controlled conditions. In this talk I will present the study of geometric structures in phase space that govern the chaotic transport in an atomic system. I will show how these results apply to understanding the chaotic ionization in Rydberg atoms. Theoretical results are based on the study of a homoclinic tangle and its corresponding turnstile. Understanding the relationship between the turnstile and the system parameters allows us to draw conclusions about the ionization process and to design the experiments for probing the structure of the chaotic phase space. Finally, I will present a set of recent results which show that this approach is valid not only in the classical regime, but also for atoms whose energy levels are in the regime frequently thought of as requiring quantum computations.