We analyze all available solar and related reactor neutrino experiments, as well as simulated future Be-7, p-p, and pep solar neutrino experiments. We treat all solar neutrino fluxes as free parameters subject to the condition that the total luminosity represented by the neutrinos equals the observed solar luminosity (the 'luminosity constraint'). Existing experiments plus the luminosity constraint show that the p-p solar neutrino flux is 1.02 +/- 0.02 (1sigma) times the flux predicted by the BP00 standard solar model; the 7Be neutrino flux is 0.93(-0.63)(+0.25) the predicted flux; and the B-8 flux is 1.01 +/- 0.04 the predicted flux. The CNO fluxes are very poorly determined. The neutrino oscillation parameters are: Deltam(2) = 7.3(-0.6)(+-0.4) x 10(-5) eV(2) and tan(2)theta(12) = 0.41 +/- 0.04. We evaluate how accurate future experiments must be to determine more precisely neutrino oscillation parameters and solar neutrino fluxes, and to elucidate the transition from vacuum-dominated to matter-dominated oscillations at low energies. A future Be-7 v-e scattering experiment accurate to +/-10% can reduce the uncertainty in the experimentally determined Be-7 neutrino flux by a factor of four and the uncertainty in the p-p neutrino flux by a factor of 2.5 (to +/-0.8%). A future p-p experiment must be accurate to better than +/-3% to shrink the uncertainty in tan(2)theta(12) by more than 15%. The idea that the Sun shines because of nuclear fusion reactions can be tested accurately by comparing the observed photon luminosity of the Sun with the luminosity inferred from measurements of solar neutrino fluxes. Based upon quantitative analyses of present and simulated future experiments, we answer the question: Why perform low-energy solar neutrino experiments?