A hydrodynamic theory for the flow of a viscoelastic fluid is formulated and applied. The viscoelasticity is specified in terms of the conventional and well‐known relaxation and retardation spectra. Time dependent recoverable strains are defined with respect to convected rotating coordinate axes. The theory is linear to the extent that these strains are small although the over‐all displacement gradients of the fluid may be large. The theory is applied to two‐dimensional steady‐state simple shear flow to predict the variation of the apparent viscosity and the steady‐state elastic compliance with shear rate. Pressure flow through a pipe of circular cross section has also been treated.Previous work of a similar nature is reviewed.