One of the principal challenges facing all implementations of quantum computing is protection of qubits from the debilitating effects of decoherence. Through unwanted interactions with the host environment of the qubit information stored in local degrees of freedom (e.g. spin in a semiconductor quantum dot) is lost. Topological quantum computing seeks to mitigate or wholly overcome decoherence by utilizing non-local quantum degrees of freedom to represent a qubit state. These non-local topological degrees of freedom do not couple to local perturbations and are consequently less susceptible to decoherence. Devising and constructing suitable physical platforms for topological qubits poses challenge for materials science. Topological degrees of freedom known as Majorana zero modes (MZMs) are predicted to form in hybrid systems consisting of low dimensional semiconductors with strong spin-orbit coupling that have been proximitized by an s-wave superconductor and placed in an appropriate magnetic field to open a topological gap. In this way the hybrid system mimics a topological p-wave spin less superconductor. A nanowire constructed from this system should support MZM at its ends. In this talk I will describe our efforts to realize MZM in hybrid systems consisting of a two-dimensional electron gas (2DEG) in strong spin-orbit coupled InAs or InSb that is proximitized with an ordinary s-wave superconductor, typically aluminum. Structures are grown in a multi-chamber molecular beam epitaxy system. We present experimental evidence for the formation of MZMs in nanowires defined through top-down lithography in the Al-InAs 2DEG system. Important materials challenges including epitaxial control of the superconductor/semiconductor interface, control of charge transfer across the interface by bandgap engineering, control of the induced superconducting gap, and reduction of disorder in a near-surface 2DEG will be discussed. New results and the impact of material parameters on device performance will be emphasized.