The possibility of using tapered or parabolic capillaries for measurement of liquid viscosities is investigated both experimentally and theoretically. It is demonstrated that even small deviations in capillary radius from a constant value may substantially affect measurement results. Equations are derived which allow correct analysis of the measurement results in tapered or parabolic capillaries. The following cases are analyzed: a water imbibition into a tapered or parabolic capillary and displacement of one liquid by another immiscible liquid in tapered or parabolic capillaries, Two possibilities are considered: (a) the narrow end of the capillary as capillary inlet and (b) the wide end of the capillary as capillary inlet.