Qubit connectivity is an important property of a quantum processor, with an ideal processor having random access -- the ability of arbitrary qubit pairs to interact directly. We describe the implementation of a random access superconducting quantum information processor using multimode circuit quantum electrodynamics, demonstrating universal quantum gate operations on a nine-bit quantum memory. The memory uses the eigenmodes of a linear array of coupled superconducting resonators, with the qubits being superpositions of vacuum and single-photon states stored in them. These qubits are all controlled by a single superconducting transmon circuit coupled to the edge of the array, serving as the central processor. We show that single transmon charge control, and flux-driven sideband interactions with the cavity modes are sufficient for universal quantum control of the entire multimode manifold, demonstrating universal gate operations between arbitrary pairs of modes, and efficient schemes for generating multi-photon entangled states. I will describe recent efforts to realize this quantum computing architecture using high-Q 3D multimode superconducting microwave cavities with photon lifetimes exceeding a millisecond. I will additionally describe methods for enhancing photon lifetimes in a superconducting circuits by engineering transition matrix elements, and Raman schemes for realizing fast gate operations.