The mathematical simulation of blast furnace operation, which uses theories of reaction kinetics and transport phenomena, has been being developed in past four decades. Although blast furnace was once treated as a black-box-reactor, a lot of experimental and numerical efforts have revealed that multiple regions exist within a furnace. The requirements for more detailed information of their functions and characteristics has been increased, as they revealed. Such demands improves the mathematical simulation model of blast furnace from one-dimensional to multi-dimensional, from steady to transient, and from single phase to multiple phases. This paper explains the outline of the latest mathematical model of blast furnace operation, which is based on multi-fluid theories, reaction kinetics and transport phenomena, and its applications. This model treats four materials, that have different flow characteristics and thermo-physical properties, as fluid. These materials are gas (blast and reaction gases), lump solids (coke and ore), liquids (molten metal and slag) and fine particles (unburnt pulverized coal and fine coke). The equations of motion of these four phases have the same form and are solved by same technique. This way of modeling allows efficient process simulation including heat and mass transfer, and reaction analyses. The model has been applied to a variety of operating conditions, and revealed the in-furnace status of blast furnace in detail.