The principles of capillary viscometry are reviewed, and design criteria for capillary viscometers to be used with non-Newtonian liquids are discussed. A description is given of a new type of variable-shear capillary viscometer—a modification of the Ubbelohde suspended-level viscometer—designed for measurements in the low-shear range. The instrument is rugged, convenient, and precise, and makes possible measurements at free fall over a 10-fold range of shear stress (or shear rate), a range that can easily be extended by use of two or more suitably modified instruments or by the application of external pressure. Experimental evidence is presented to show that proper choice of the dimensions of the instrument eliminates or minimizes most of the "effects" that limit the precision of measurements of relative viscosity (and related functions) of high polymer solutions in the range of low shear stresses.