After a short survey of the physics of solar neutrinos, giving an overview of hydrogen burning reactions, predictions of standard solar models and results of solar neutrino experiments, we discuss the solar-model-independent indications in favour of non-standard neutrino properties. The experimental results appear to be in contradiction with each other, even after disregarding some experiments: unless electron neutrinos disappear in their trip from the sun to the earth, the fluxes of intermediate energy neutrinos (those from Be-7 electron capture and from the CNO cycle) turn out to be unphysically negative, or extremely reduced with respect to standard solar model predictions. Next we review extensively the nonstandard solar models built as attempts to solve the solar neutrino puzzle. The dependence of the central solar temperature on chemical composition, opacity, age and on the values of the astrophysical S-factors for hydrogen-burning reactions is carefully investigated. Also, possible modifications of the branching among the various pp-chains in view of nuclear physics uncertainties are examined. Assuming standard neutrinos, all solar models examined fail in reconciling theory with experiments, even when the physical and chemical inputs are radically changed with respect to present knowledge and even if some of the experimental results are discarded.