What does finding periodic orbits in the 3-body problem, inscribing a square in a smooth curve, finding knot invariants, and the formation of rainbows, share in common? A possible answer is that all these apparently unrelated problems can be understood and solved by studying Lagrangian submanifolds, a particular type of submanifolds prevalent in symplectic topology. In this talk, we will first introduce and motivate Lagrangians and their intersection theory, focusing on the surface case. Then, we will discuss recent developments on their classification, including a surprising connection to cluster algebras.