We shall give the geometric definition of several holomorphic curve invariants for symplectic and contact manifolds, such as contact homology, symplectic homology and some of their variants. We shall then explain the relationship between these theories, leading to a common algebraic framework. These relations can also be used to translate structural results from one of these invariants to the other ones. This can be illustrated with the effect of some geometric operations on these invariants, such as Legendrian surgery.