Introduction
Lecture 1 - Symplectic manifolds and lagrangian submanifolds, examples
Lecture 2 - Lagrangian splittings, real and complex polarizations, Kähler manifolds
Lecture 3 - Reduction, the calculus of canonical relations, intermediate polarizations
Lecture 4 - Hamiltonian systems and group actions on symplectic manifolds
Lecture 5 - Normal forms
Lecture 6 - Lagrangian submanifolds and families of functions
Lecture 7 - Intersection Theory of Lagrangian submanifolds
Lecture 8 - Quantization on cotangent bundles
Lecture 9 - Quantization and polarizations
Lecture 10 - Quantizing Lagrangian submanifolds and subspaces, construction of the Maslov bundle

The first six sections of these notes contain a description of some of the basic constructions and results on symplectic manifolds and lagrangian submanifolds. Section 7, on intersections of largrangian submanifolds, is still mostly internal to symplectic geometry, but it contains some applications to machanics and dynamical systems. Sections 8, 9, and 10 are devoted to various aspects of the quantization problem. In Section 10 there is a feedback of ideas from quantization theory into symplectic geometry itslef.