We introduce the notion of a multi-fan. It is a generalization of that of a
fan in the theory of toric variety in algebraic geometry. Roughly speaking a
toric variety is an algebraic variety with an action of algebraic torus of the
same dimension as that of the variety, and a fan is a combinatorial object
associated with the toric variety. Algebro-geometric properties of the toric
variety can be described in terms of the associated fan. We develop a
combinatorial theory of multi-fans and define ``topological invariants'' of a
multi-fan. A smooth manifold with an action of a compact torus of half the
dimension of the manifold and with some orientation data is called a torus
manifold. We associate a multi-fan with a torus manifold, and apply the
combinatorial theory to describe topological invariants of the torus manifold.
A similar theory is also given for torus orbifolds. As a related subject a
generalization of the Ehrhart polynomial concerning the number of lattice
points in a convex polytope is discussed
