"Entanglement swapping describes a process by which two particles with no common past can be entangled with one another. Considered a key component of quantum repeater protocols, the method of entanglement swapping is also used for creating multipartite entangled states, such as Greenberger-Horne-Zeilinger (GHZ) states. To date, experimental implementations of entanglement swapping and multipartite entanglement have focused on two-dimensional or qubit entanglement. This is primarily because such experiments are usually performed with polarization-entangled photons, which reside in a two-dimensional Hilbert space. On the other hand, photons entangled in orbital angular momentum (OAM) live in a discrete, infinite-dimensional Hilbert space. Here, I propose a method for creating high-dimensional GHZ states entangled in OAM. The use of a polarizing beam splitter is key for swapping bipartite polarization entanglement to create GHZ states entangled in polarization. With this in mind, the first goal of my proposal is to design the equivalent of a polarizing beam splitter for OAM, which separates and recombines photons based on their OAM. The second part of my proposal focuses on using this device for a high-dimensional multi-stage OAM-entanglement swapping experiment, and exploring its applications in quantum repeater protocols. In the third part, I propose two different methods of creating quadripartite OAM-GHZ states by swapping OAM and polarization-entanglement among four bipartite states. High-dimensional multipartite entangled states have the potential to enable vastly more efficient methods of quantum computing and greatly enhance the security of quantum cryptographic protocols. Additionally, such states are considered essential for fundamental tests of quantum theory. An overlying goal of this proposal will be to explore the applications of OAM-GHZ states for the enhancement of quantum technologies such as superdense coding and quantum secret sharing protocols."