Abstract: The universal quantum computer is a device capable of simulating any physical system and represents a major goal for the field of quantum information science. In the context of quantum information, `universal' refers to the ability to carry out arbitrary unitary transformations in the system's computational space. Combining arbitrary single-quantum-bit (qubit) gates with an entangling two-qubit gate provides a set of gates capable of achieving universal control of any number of qubits, provided that these gates can be carried out repeatedly and between arbitrary pairs of qubits. Although gate sets have been demonstrated in several technologies, they have so far been tailored towards specific tasks, forming a small subset of all unitary operators. Here we demonstrate a quantum processor that can be programmed with 15 classical inputs to realize arbitrary unitary transformations on two qubits, which are stored in trapped atomic ions. Using quantum state and process tomography, we characterize the fidelity of our implementation for 160 randomly chosen operations. This universal control is equivalent to simulating any pairwise interaction between spin-1/2 systems. A programmable multiqubit register could form a core component of a large-scale quantum processor, and the methods used here are suitable for such a device.