Viscosity is a thermophysical property of paramount importance, being essential for many scientific and industrial applications. The most common instruments for its measurement are glass capillary viscometers. Therefore, the use of capillary viscometers is widespread both in industry and in research. The range of viscosities of interest range from lower than that of water to several orders of magnitude higher values, the measurement of which requires different capillary viscometers. Most of the practical applications concern routine instruments, mainly for quality control. One main issue for the utilization of capillary viscometers relates to the need for their calibration, assuring its traceability to the water primary viscosity standard, to certify its worldwide validity. The present paper focuses on capillary instruments dedicated to perform viscosity measurements on Newtonian organic liquids at atmospheric pressure, as it is assumed that is the most widespread type of application for these viscometers. Capillary viscometry has a completely well-defined working equation, namely, the Hagen–Poiseuille equation. However, the practical performance of the measuring instruments deviates from that working equation. Most of those deviations are currently considered by many users. However, some of those deviations have not reached that status yet, like those concerning the effects due to the surface tension of the sample on the measurements. All these aspects are summarized and analyzed in the present article, together with a brief general description of the most common types of capillary viscometers, namely, the Ostwald and the constant-level or Ubbelohde instruments.