We discuss the results of Homestake and Kamioka experiments, showing that if the results of these experiments are taken at their face values - one way to save ''conventional neutrinos'' is to look for a nuclear solution decreasing both Be-7 and B-8 neutrino fluxes with respect to the predictions of the standard solar models. Recent GALLEX results appear in agreement with such a conclusion. We discuss the sensitivity of the B-8 and Be-7 neutrino fluxes to the behaviour of the low energy He-3 + He-3 and He-3 + He-4 cross sections. We derive analytically the dependence of the neutrino fluxes on the low energy nuclear cross sections. This analytical approach has been supported by numerical experiments based on a new Standard Solar Model. In the non-resonant case, reduction of the neutrino fluxes to about 1/3 of the Standard Solar Model could be obtained if the true value of S34(0) is three times smaller than the presently accepted extrapolated value. Alternatively, one should have S33(0) wrong by a factor nine. A resonance in the He-3 + He-3 channel could yield a sufficient reduction of B-8 neutrinos and, furthermore, a suppression of Be-7 neutrinos larger than that of B-8 neutrinos provided that E(R) less-than-or-equal-to 21.4 keV, an energy region so far almost unexplored experimentally. We show that future experiments in underground laboratories should be able to explore the region down to E(R) = 10 keV with a significant sensitivity. We also compare our Standard Solar Model with the results of previous calculations.