There are many rheological problems in the melt spinning process. They can be classified into two categories, i.e., the problems in shearing flow and those in extensional flow. The former is concerned with extrusion of molten PET (polyethyleneterephthalate) through spinneret orifices. Bagley’s plots showing the relationship of pressure drop toL/Dof orifice were done to obtain the end correlation in the capillary flow of PET. By using this end correction values, the relaxation time of molten PET was estimated to be around 2×10−3s under some assumptions. With regard to the problems in the extensional flow three typical and important examples were shown from both experimental and theoretical aspects. (1) Fine denier filament spinning for obtaining fibers with less than about 10 μm in diameter is a challengeable theme in the meaning of a conflict with extensional flow instability, so called draw resonance phenomenon. To solve this difficulty, it can be suggested that optimal spinneret orifice design will be required for instance. (2) High speed spinning is a process of stress‐induced crystallization of PET. Dependence of spinning behaviors such as neck‐like deformation and mechanical/structural properties on PET molecular weight was investigated. Crystallization of higher molecular weight PET was found to occur at lower spinning speed compared with that of lower one. (3) Air jet melt spinning for multifilament system was investigated both theoretically and experimentally. Computation results showed a good agreement with experimental ones in terms of the stretching speed in ejector. Birefringence values of as‐spun fibers can be estimated by using computed outputs.