Experiments to determine the value of Taylor's longitudinal diffusivity in turbulent flows in open channels and circular pipes have produced results which are in many cases inconsistent with one another and with theoretical estimates due respectively to Elder (1959) and Taylor (1954). Neither is there general agreement on when Taylor's theory becomes applicable. In an attempt to clarify the discrepancies two well-known sets of experiments by Fischer (1966) and Taylor (1954) are re-examined by using a natural procedure which, it is argued, has certain advantages over more usual methods. It is shown that Fischer's observations in an open channel were not made at a sufficient distance downstream from the point of injection for Taylor's theory to apply but that they are consistent with a description of the early stages of the dispersion process due to Sullivan (1968). It is subsequently argued that these observations suggest that Elder's estimate of the diffusivity is too low for two reasons. The first is the error caused by assuming the existence of an eddy diffusivity calculated by means of Reynolds analogy and the second is the neglect of the viscous sublayer. On the other hand it is shown that some of Taylor's observations in a circular pipe are consistent with his theory, although the values of the diffusivity which best fit the data are about 25% higher than Taylor's estimate, and it is suggested that this is for the same reasons as in the open channel. The paper concludes with a discussion of the effect of the viscous sublayer on the value of the longitudinal diffusivity. Partly on the basis of an approximate model it is argued that theoretical calculations which ignore the viscous sublayer are too low by amounts which depend on the Reynolds and Schmidt numbers and can be of the order of 20%.