Measurement of e+e- to hadrons cross sections at BABAR, and implication for the muon g-2

The BaBar Collaboration has an intensive program of studying hadronic cross sections at low-energy $e^+e^-$ collisions, accessible via initial-state radiation. Our measurements allow significant improvements in the precision of the predicted value of the muon anomalous magnetic moment, that shed light on the current $\approx$ 3.5 sigma difference between the predicted and the experimental values. We have published results on a number of processes with two to six hadrons in the final state. We report here the results of recent studies with the final states that constitute the main contribution to the hadronic cross section below 3 GeV, as $e^+e^- \rightarrow \pi^+\pi^-$, $K^+K^-$, $K_S K_L$ and $e^+e^- \rightarrow 4$ hadrons.