Purely elastic flow instabilities-those occurring in the absence of in
ertial forces-in flows used for the measurement of fluid viscometric p
arameters are reviewed. In particular, we examine the growing body of
knowledge surrounding how the presence of elasticity in homogeneous, f
lowing polymeric fluids can act in isolation to create flow bifurcatio
n and change. The field of inquiry is relatively new, and the studies
that have played a key role in developing the science include a balanc
e of experiment, large-scale computer simulation, and analytic stabili
ty theory. All are reviewed with a focus on how each has contributed t
o the understanding of novel physical mechanisms and principles govern
ing these instabilities. The flows examined are simple, but critically
important to the measurement devices common to the laboratories of rh
eologists and fluids engineers. Moreover, it is clear that the knowled
ge gained from these studies may be generally applicable to a wide ran
ge of much more complicated flows that demonstrate this type of instab
ility.