In order to provide fundamental bases for incorporating the effects of favourable streamwise pressure gradients into models for internal turbulent flows and turbulent boundary layers, time series measurements with a cross-wire probe and a wall shear stress sensor were obtained simultaneously in an oil channel both for fully developed and laterally converging flows. Data were concentrated in the viscous layer and at the centreplane for slight to highly favourable pressure gradients. (Here the viscous layer is defined as the region where viscous effects are significant, but not necessarily dominant; it includes the ‘laminar’ and buffer sublayers in the terminology of some investigators). Results presented include comparisons of the profiles of the mean statistics, plus correlations and spectra, of streamwise and wall-normal components, their product and the wall shear stress. The key new data are the measurements of the fluctuating normal component and related statistics in the viscous layer for highly favourable pressure gradients. For the outer half of the viscous layer its root-mean-square fluctuations decrease as the pressure gradient is increased, consistent with heretofore unconfirmed predictions from direct numerical simulations. Based on examination of the probability density distributions, one may conclude that an effect of a strong pressure gradient is to reduce transport of momentum in the outer part of the viscous layer.