In this paper, translated and consistent versions of the Peng-Robinson (tc-PR) and Redlich-Kwong (tc-RK) cubic equations of state(CEoS) are developed. The adjective consistent means that the alpha-function used in such models passes the consistency test we recently developed and which guarantees safe extrapolation in the supercritical region and safe VLE calculation in multi-component systems. The adjective translated means that a volume translation aimed at exactly reproducing the experimental saturated liquid volume at a reduced temperature of 0.8 was used. The key conclusion of this paper is that the tc-PR EoS is certainly the safest and the most accurate 3-parameter cubic EoS ever published. Indeed the average deviations over roughly 1000 compounds belonging to different chemical families are: Delta P-sat<1 %, Delta H-vap = Delta c(P,L)(sat) = 2 % and Delta v(L)(sat) (T-r < 0.9) = 2.3 %. In the case of a lack of experimental data to fit the 3 parameters of the alpha-function and to determine the value of the volume correction, the second part of this study aims at providing a generalized version of the tc-PR and tc-RK CEoS in which all parameters can be estimated from the mere knowledge of the acentric factor. (C) 2016 Elsevier B.V. All rights reserved.