This is joint work with Catherine Wolfram. We describe a new family of determinantal probability measures in d dimensions called Independent Coverings. Given a region G in Z^d of size n, and fixed polyomino t, an independent covering of G by t is collection of n different translates of t which covers G and is independent in the sense that the incidence matrix between those translates and G is of full rank.  We define a natural probability measure on the set of independent coverings, show that it is a determinantal measure, and compute some of its properties, including scaling limits and, sometimes, conformal invariance.   Generalizations to multiple tiles are readily available.