The computation of the grass reference evapotranspiration with the FAO56 Penman-Monteith equation (PM-ET o ) requires data on maximum and minimum air temperatures (T max , T min ), actual vapour pressure (e a ), shortwave solar radiation (R s ), and wind speed at 2 m height (u 2 ). Nonetheless, related datasets are often not available, are incomplete, or have uncertain quality. To overcome these limitations, several alternatives were considered in FAO56, while many other procedures were tested and proposed in very numerous papers. The present study reviews the computational procedures relative to predicting the missing variables from temperature, i.e., the PM temperature approach (PMT), and estimating ET o with the Hargreaves-Samani (HS) equation. For the PMT approach, procedures refer to predicting: (a) the dew point temperature (T dew ) from the minimum or the mean air temperature; (b) shortwave solar radiation (R s ) from the air temperature difference (TD = T max -T min ) combined with a calibrated radiation adjustment coefficient (k Rs ); and (c) wind speed (u 2 ) using a default value or a regional or local average. The adequateness of computing T dew from air temperature was reassessed and the preference for using an average u 2 has been defined. To ease the estimation of R s, for the PMT approach and the coefficient of the HS equation, multiple linear regression equations for predicting k Rs were developed using local averages of the temperature difference (TD), relative humidity (RH) and wind speed as independent variables. All variables were obtained from the Mediterranean set of CLIMWAT climatic data. Two types of k Rs equations were developed: climate-focused equations specific to four climate types -humid, sub-humid, semi-arid, and hyper-arid and arid -, and a global one, applicable to any type of climate. The usability of the k Rs equations for the PMT and HS methods was assessed with independent data sets from Bolivia, Inner Mongolia, Iran, Portugal and Spain, covering a variety of climates, from hyper-arid to humid. With this purpose, ET o estimated with PMT and HS (ET o PMT and ET o HS ) were compared with PM-ET o computed with full data sets to evaluate the usability of the k Rs equations. Adopting the climate-focused k Rs equations with ET o PMT , the RMSE averaged 0.59, 0.64, 0.65 and 0.72 mm d −1 for humid, sub-humid, semi-arid, and arid and hyper-arid climates, respectively, while the RMSE values relative to ET o HS when using the respective climate-focused k Rs equations averaged 0.58, 0.60, 0.60 and 0.69 mm d −1 for the same climates. These results are similar to those obtained with the k Rs global equation. The accuracy of the PMT approach when using the k Rs equations was also evaluated when one, two, or all three T dew , R s and u 2 variables are missing and the resulting goodness-of-fit indicators demonstrated the advantage of the combined use of observed and estimated weather variables. The usability of the k Rs equations for an efficient parameterization of both the PMT approach and ...