  Random numbers are required for a variety of applications from secure
communications to Monte-Carlo simulation. Yet randomness is an asymptotic
property and no output string generated by a physical device can be strictly
proven to be random. We report an experimental realization of a quantum random
number generator (QRNG) with randomness certified by quantum contextuality and
the Kochen-Specker theorem. The certification is not performed in a
device-independent way but through a rigorous theoretical proof of each outcome
being value-indefinite even in the presence of experimental imperfections. The
analysis of the generated data confirms the incomputable nature of our QRNG.
