After controlling the composition and temperature, and removing nonmetallic inclusions, the molten steel is transferred in a ladle and poured into a mold, where it solidifies to produce semi-finished or finished products. In the past, the ingot casting-rolling (slabbing, blooming, or billeting) process was commonly used. In this process, the molten steel was poured into many cast-iron ingot molds and, when the solidification was complete, the ingots were taken out, reheated, and rolled by a slabbing, blooming, or billeting mill. The continuous casting process has now virtually replaced this earlier method. In continuous casting, the molten steel in the ladle is poured into an intermediate vessel(tundish), released into a hollow water-cooled copper mold, and continuously withdrawn from the bottom of the mold as a shell begins to form around the molten metal. The reasons for this change include: (i) the reheating and slabbing process can be omitted because the cast strand has a near-net shape similar to that of the semi-finished product; (ii) the yield is much higher because the continuously cast strand has only two small end portions, in contrast to the tops and bottoms which must be cropped from every ingot; (iii) solute element segregation and nonmetallic inclusions are much lower; and (iv) advanced technologies have improved the productivity and surface quality of the cast pieces greatly, to such an extent that productivity has become compatible with that of the converter and hot rolling processes, thus providing balanced continuity among these processes. The continuous caster allows a cast strand to be withdrawn at high speed (1.5-2.8 m/min) from the mold in the form of a core of molten steel encased by a thin solidified shell. This high withdrawal speed ensures that casting productivity is matched to that of the converter. As the cast strand descends from the mold, its surface is cooled by a water spray or water mist, and the thickness of the shell increases progressively as the material solidifies. However, the ferrostatic pressure of the molten steel rises at the same time. The cast strand is therefore supported by rolls so that the solidified shell does not bulge. If the solidified shell is deformed due to thermal strains or ferrostatic pressure, cracks form on both the surface and in the interior due to the low ductility and low strength of the shell at high temperatures. An analysis of heat transfer between the molten steel/solidified shell/mold or spray is necessary to increase productivity and prevent deformation and cracking. In addition to this analysis, it is imperative to analyze stress, strain, and deformation in the solidified shell when it passes through both the mold and the support rolls. Progress has been made in the analyses of the heat transfer, elastic-plastic thermal stress, and creep-behavior of the cast strand by use of the finite difference and finite element method, and various computational programs simulating these phenomena have been developed. The measurement of the dynamic behavior of steel at elevated temperatures necessary for such computations has also been carried out. In addition to deformation and cracking, the quality of the cast strand is impaired by the presence of nonmetallic inclusions and by the segregation of solute elements. Owing to the reoxidation of the molten steel by air, entrainment of slag and refractories, etc., the number of nonmetallic inclusions increases as the steel moves progressively from the ladle to the tundish to the mold. To minimize this problem, the flow of molten steel within the tundish and through the nozzle between the tundish and mold is carefully controlled to ensure coagulation, flotation, and separation of nonmetallic inclusions. Progress has been made in research to evaluate this flow of molten steel by simulation tests with water models and by mathematical modeling of fluid dynamics based on numerical solutions to the governing differential equations, including turbulent forms of the Navier-Stokes equation. The solubility of solute elements is usually lower in the solid state than in molten steel. These solute elements are discharged into the molten steel at the front face of columnar dendrites of the solidified shell which grows as solidification proceeds. These solute elements concentrate, resulting in positive segregation. The segregation of carbon is shown schematically in the right hand side of the figure. Strong segregation occurs during the final stage of solidification between the branches of columnar dendrites and also at the center-thickness of the cast strand. Solidification theories have been established for the relationship between the morphology of growing crystals and the temperature gradient and cooling rate, the segregation of solute elements near the front of solidifying shell, the rate of solidification which affects segregation, and the influence of the flow of molten steel.