After the seminal work of Paul Rabinowitz on periodic orbits of Hamiltonian Systems on starshaped surfaces in |R^n, Contact Structures have become a natural object of study for analysts. The search for invariants for these contact forms/structures benefited very much from the deeper understanding of the much more general associated variational problem used in the work of Paul Rabinowitz and of Conley-Zehnder. Contact Homology has then been defined using pseudo-holomorphic curves, but also via Legendrian curves. After broadly recalling the main steps in the formulation and the development of these tools, we present a more detailed account of the contact homology via Legendrian curves, including its definition, its compactness properties and the value of this homology for odd indexes.