By contrast to polyethylene terephthalate (PET), extrusion foaming of polybutylene terephthalate (PBT) to medium-low densities has been seldom reported in the literature. In this study, a commercial linear PBT resin was reactively modified in a batch mixer to a branched structure with a higher molecular weight (MW) and a broader molecular weight distribution (MWD) as evidenced by rheological analysis. Chain branching was also accomplished by single screw extrusion, where the competing degradation reaction needed to be taken into account. Optimization of the extrusion operational conditions, which also involved the use of suitable consititutive equations, led to the production of a branched resin with viscoelastic characteristics suitable for low density extrusion foaming by injection of physical blowing agents (PBA). The branched product made under the optimized conditions showed good foamability. The effect of pressure drop rate on cell nucleation rate and the effect of the crystallization rate on cell density and cell morphology are examined.