Symplectic geometry originates from the theory of Hamiltonian dynamical systems in classical physics. The phase space of a dynamical system can be formalized as the cotangent bundle of a smooth manifold, which carries a canonical symplectic structure. All the key concepts of Hamiltonian dynamics turn out to be natural geometric objects for this structure.In the 1980s Vladimir Arnol'd formulated the fundamental Nearby Lagrangian Conjecture: every closed exact Lagrangian submanifold of the cotangent bundle of a closed manifold is Hamiltonian isotopic to the zero section.This conjecture is still wide open and plays a central role in the modern development of symplectic topology. Our aim is to motivate this question and give a flavor of the ideas and techniques that lead to some recent progresses.