Contact homology is a powerful invariant of contact structures,introduced by Eliashberg,Givental and Hofer [6] in the bigger framework of Symplectic Field Theory.In this mini-course we will give a short introduction to contact homology,focusing on applications to Hamiltonian dynamics.The applications include the existence and multiplicity of periodic orbits for Reeb flows and some results concerning the stability of these orbits.In particular,we will discuss the notion of symplectically degenerate maximum(SDM)and explain how Ginzburgs celebrated proof of the Conley conjecture [7] can be adapted to the contact setting to show that the existence of a simple SDM forces the Reeb flow to have infinitely many periodic orbits [8].This was used to give a new proof of a recent result due to Cristofaro-Gardiner and Hutchings [5] on the existence of at least two periodic orbits for every Reeb flow on the standard contact three-sphere.Many of the applications that will be discussed are based on joint works with M.Abreu,V.Ginzburg,D.Hein,U.Hryniewicz and G.Paternain.