Contact homology is a Floer-type invariant for contact manifolds, and is a part of Symplectic Field Theory. One of its first applications was the existence of exotic contact structures on spheres. Originally, contact homology was defined only for closed contact manifolds. We will describe how to extend it to open contact manifolds that are "convex". As an application, we prove the existence of (infinitely many) exotic contact structures on $\mathbb{R}^{2n+1}$ for all $n>1$.This is joint with François-Simon Fauteux-Chapleau.