In this talk, we aim to outline a proof of the Poincaré-Birkhoff fixed point theorem which states that any area preserving diffeomorphism of the annulus which preserves the two boundary components and twists them in opposite directions has at least two fixed points. We will also discuss its application to Hamiltonian systems and other fixed point theorems (for higher dimensions) in symplectic geomtery if time permits. (EH 2866)