One of the most important events in science dates back to 1687, when Newton published the Philosophiæ Naturalis Principia Mathematica. In this masterpiece of human thought, the famous second law of motion is laid out, which concretely and beautifully expresses the relationship between the motion of a classical particle and the forces exerted on it. This work contains the solution to the two-body problem, concerning the motion of two massive bodies under gravitational interaction. However, when a third mass is added, known as the three-body problem, there is no such concrete solution, as Henri Poincaré exhibited in 1892. Since then, the role of celestial mechanics has been central to modern scientific discourse, in view of its connections with astronomy and space exploration. Symplectic geometry, the geometry underlying William Rowan Hamilton's reformulation of classical mechanics, serves as the modern mathematical framework instrumental to addressing these classical problems. This talk will attempt to thread together bits and pieces of this beautiful story, from the (inevitably biased) viewpoint of a modern practitioner.     Agustin Moreno is a Junior Professor in the Department of Mathematics at Heidelberg University in Germany, as well as a Member of the School of Mathematics at IAS. He holds degrees from the University of Cambridge, where he completed part III of the Mathematical Tripos with distinction (and was awarded for his distinguished performance), and from the Humboldt University in Berlin, where he finished his Ph.D. (started at University College London, UK). He has been affiliated with the Berlin Mathematical School, and has been a Research Fellow at the Mittag-Leffler Institute in Sweden, an institution famously tied to the work of Poincaré on the three-body problem, where much of his work on the very same problem was carried out. His research lies at the crossroads of geometry, topology, dynamics and mathematical physics, with further interest in applications to space mission design, which he pursues in collaboration with navigational engineers at NASA.