The harmonization of international temperature measurements requires the high-accuracy realization of many different temperature reference points. This results from the feature of the intensive measurand temperature that temperatures cannot simply be divided or multiplied. Thus, the points must cover the whole range of interest, at present from 1 mK to a few 1000 K. Furthermore, instruments are necessary for the interpolation between the non-continuous guide values. This led to the establishment of International Temperature Scales (ITS). The ITS prescribe interpolation instruments and assign fixed temperature values to suitable phase transitions without uncertainty. The large temperature range can only be covered by applying very different phase transitions. This includes the classical transitions, namely triple, melting, and freezing points, but also second-order transitions, as superfluid and superconducting ones, and the very new eutectic or peritectic points of metal-carbon compositions. A high-accuracy realization requires a reliable uncertainty estimation. This is, therefore, the central topic of this review. Since a given non-ideal condition of a sample, especially the impurity content, cannot be reproduced as accurate as necessary, the fixed- and reference-point temperatures are defined for ideal substances under ideal conditions. Thus, the estimation of the uncertainty of the realizations must be based on estimating the magnitude of all physical effects influencing the observed phase-transition temperature. The application of this methodology is discussed in the paper as unifying topic independent of the individual problems to be solved. Furthermore, recommendations of the Consultative Committee for Thermometry are summarized, and own experiences are supplemented.