  We discuss the treatment of quantum-gravitational fluctuations in the
space-time background as an `environment', using the formalism for open
quantum-mechanical systems, which leads to a microscopic arrow of time. After
reviewing briefly the open-system formalism, and the motivations for treating
quantum gravity as an `environment', we present an example from general
relativity and a general framework based on non-critical strings, with a
Liouville field that we identify with time. We illustrate this approach with
calculations in the contexts of two-dimensional models and $D$ branes. Finally,
some prospects for observational tests of these ideas are mentioned.
