  We introduce the notion of a topological toric manifold and a topological fan
and show that there is a bijection between omnioriented topological toric
manifolds and complete non-singular topological fans. A topological toric
manifold is a topological analogue of a toric manifold and the family of
topological toric manifolds is much larger than that of toric manifolds. A
topological fan is a combinatorial object generalizing the notion of a
simplicial fan in toric geometry.
  Prior to this paper, two topological analogues of a toric manifold have been
introduced. One is a quasitoric manifold and the other is a torus manifold. One
major difference between the previous notions and topological toric manifolds
is that the former support a smooth action of an $S^1$-torus while the latter
support a smooth action of a $\C^*$-torus. We also discuss their relation in
details.
