Lagrangian submanifolds appear naturally in symplectic
geometry from different angles: real algebraic geometry, dynamics and
integrable systems are just a few sources to mention. In fact, once
the concept is introduced many symplectic phenomena can be rephrased
in a Lagrangian language, as a famous citation of A. Weinstein goes
"Everything is Lagrangian".
I will survey the development of Lagrangian topology, which is a
theory that mixes topological and symplectic invariants associated to
Lagrangian submanifolds. I will explain how these can be applied to
solve various problems in symplectic geometry. I will then move on to
newer developments that have to do with the algebraic structure of the
totality of these invariants, such as the Donaldson and Fukaya
categories. If time permits I will outline a new and more geometric
approach to study these categories.
No apriori knowledge of symplectic geometry is needed.