I will give an introduction to symplectic geometry and Hamiltonian systems and then introduce an invariant called symplectic cohomology. This has many applications in symplectic geometry and has been used a lot especially in the last 5-10 years. I will explain some basic properties of this invariant and how it can be used to find periodic orbits of a Hamiltonian system. I will also explain how this invariant can be used to distinguish symplectic manifolds