  Generalizing the passage from a fan to a toric variety, we provide a
combinatorial approach to construct arbitrary effective torus actions on
normal, algebraic varieties. Based on the notion of a ``proper polyhedral
divisor'' introduced in earlier work, we develop the concept of a ``divisorial
fan'' and show that these objects encode the equivariant gluing of affine
varieties with torus action. We characterize separateness and completeness of
the resulting varieties in terms of divisorial fans, and we study examples like
C*-surfaces and projectivizations of (non-split) vector bundles over toric
varieties.
