A stochastic general relativity is derived via compactification of (n + v)-dimensional Kaluza–Klein gravity, or effective superstring theory, on an internal space (a v-torus Tv = S1 ⊗ S1 ⊗ ... ⊗ S1) whose volume is parametrised by a random Gaussian modulus (scalar) field. The dimensional reduction, Mn + v → Mn ⊗ Tv, leads to stochastic vacuum Einstein equations on Mn, with a source term arising from random fluctuations or 'turbulence' in geometry on the order of the compactification scale. The source plays the role of a stochastic cosmological constant and the new Einstein vacuum equations still obey the Bianchi identities. The methodology attempts to extend the smooth manifold paradigm of general relativity—in lieu of any existing theory of quantum gravity—to accommodate short distance stochastic structure near the Planck scale. An equivalent description, and interpretation, is possible in terms of random conformal metric fluctuations in n-dimensions: given the usual vacuum Einstein equations, for (Mn, g), there exists conformal metric fluctuations Ω2g, such that the stochastic vacuum equations—derived previously via the dimensional reduction—are reproduced on (M, Ω2g). The stochastic extension of general relativity is tentatively applied to a number of key issues and scenarios of interest. These include geodesic focusing, completeness and conjugate points; energy conditions, singularities and global structure; and gravitational collapse and cosmology.