  We provide a construction of Roe (C*-)algebras of general coarse spaces in
terms of coarse geometric modules. This extends the classical theory of Roe
algebras of metric spaces and gives a unified framework to deal with either
uniform or non-uniform Roe algebras, algebras of operators of controlled
propagation, and algebras of quasi-local operators; both in the metric and
general coarse geometric setting. The key new definitions are that of coarse
geometric module and coarse support of operators between coarse geometric
modules. These let us construct natural bridges between coarse geometry and
operator algebras.
  We then study the general structure of Roe-like algebras, and investigate
several structural properties, such as admitting Cartan subalgebras or
computing their intersection with the compact operators. Lastly, we prove that
assigning to a coarse space the K-theory groups of its Roe algebra(s) is a
natural functorial operation.
