  We propose a new method to define theories of random geometries, using an
explicit and simple map between metrics and large hermitian matrices. We
outline some of the many possible applications of the formalism. For example, a
background-independent measure on the space of metrics can be easily
constructed from first principles. Our framework suggests the relevance of a
new gravitational effective action and we show that it occurs when coupling the
massive scalar field to two-dimensional gravity. This yields new types of
quantum gravity models generalizing the standard Liouville case.
