We propose a quantum device that can approximate any projective measurement on a qubit-a quantum "multimeter." The desired measurement basis is selected by the quantum state of a "program register." Two different kinds of programs are considered and in both cases the device is optimized with respect to maximal average fidelity (assuming uniform distribution of measurement bases). Quantum multimeters exhibiting the covariance property are introduced and an optimal covariant multimeter with a single-qubit program register is found. Possible experimental realization of the simplest proposed device is presented.