We re-egzamine the possible magnitude of the supersymmetric contribution to R b with imposed all available phenomenological constraints and demanding good quality of the global fit to the precision electroweak data. For low tan β we find a new region of the parameter space, with M 2 ≈ |µ| and µ < 0 where R b remains large, ∼ 0.2180 even for the lighter chargino as heavy as 90 − 100 GeV. It is an interesting mixture of the up-higgsino and gaugino. The rôle of various phenomenological constraints is discussed in analitic form and importance of small but non-negligible left-right mixing in the stop sector is emphasized in this context. The large tan β option for enhancement of R b is also reviewed. The available data do not rule out this scenario. A considerable excitement has recently been inspired by the R b and R c anomaly [1,2,3,4,5,6,7,8,9]. The succesful tests of the Standard Model (SM), to a per mille level [10], are challenged by the measurements of the partial widths of the Z 0 decays into bb and cc quarks which disagree with the SM predictions for m t = 180(170) GeV at the level of 3.7(3.5) and 2.5(2.5) standard deviations, respectively [11]. If both results are confirmed, the SM and its simplest supersymmetric extension, the Minimal Supersymmetric Standard Model (MSSM), are ruled out. However, since the R b anomaly is statistically more significant, it is also of interest to discuss the possibility of explaining only the larger than in the SM value of R b . Even if R c is fixed to its SM prediction, R c = 0.172 the then measured value of R b = 0.2206 ± 0.0016 is still 3 standard deviations away from its SM value. The issue has been addressed in particular in the framework of the MSSM [2,3,4,5,6,7,8,9]. It is well known already for some time that in the MSSM there are new contributions to the Z 0 bb vertex which can significantly enhance the value of R b (but not R c ) if some superpartners are sufficiently light [12,2,4,5,6,8,9]. More specifically, for low (large) tan β the dominant contributions are chargino-stop (CP −odd Higgs boson and chargino-stop) loops.Any improvement in R b must not destroy the perfect agreement of the SM with the other precision LEP measurements and must be consistent with several other experimental constraints (which will be listed later on). It is, therefore, important to discuss the changes in R b in the context of global fits to the electroweak data (and with all additional constraints included). Such fits in the effective low energy MSSM (unconstrained by any GUT assumptions about the pattern of soft supersymmetry breaking scalar masses) have shown that it is realistic to obtain the values of R b up to R b = 0.2180(0.2190) for small (large) tan β values [5]. Although still away by 1.5σ (1.0σ) even from the central experimental value obtained with R c fixed to its SM value, those results provide an interesting improvement over the SM prediction R b = 0.2158 (0.2160) for m t = 180 (170) GeV. At the same time the overall best χ 2 is smaller than in the SM fits by ∆χ 2 ...