The physics of nuclear reactions in stellar plasma is reviewed with special emphasis on the importance of the velocity distribution of ions. Then the properties (density and temperature) of the weak-coupled solar plasma are analysed, showing that the ion velocities should deviate from the Maxwellian distribution and could be better described by a weakly-nonextensive (|q-1| < 0.02) Tsallis' distribution. We discuss concrete physical frameworks for calculating this deviation: the introduction of higher-order corrections to the diffusion and friction coefficients in the Fokker-Planck equation, the influence of the electric-microfield stochastic distribution on the particle dynamics, a velocity correlation function with long-time memory arising from the coupling of the collective and individual degrees of freedom. Finally, we study the effects of such deviations on stellar nuclear rates, on the solar neutrino fluxes, and on the pp neutrino energy spectrum, and analyse the consequences for the solar neutrino problem.