  Random non-commutative geometries are introduced by integrating over the
space of Dirac operators that form a spectral triple with a fixed algebra and
Hilbert space. The cases with the simplest types of Clifford algebra are
investigated using Monte Carlo simulations to compute the integrals. Various
qualitatively different types of behaviour of these random Dirac operators are
exhibited. Some features are explained in terms of the theory of random
matrices but other phenomena remain mysterious. Some of the models with a
quartic action of symmetry-breaking type display a phase transition. Close to
the phase transition the spectrum of a typical Dirac operator shows
manifold-like behaviour for the eigenvalues below a cut-off scale.
