The BNABAR Collaboration has an intensive program studying hadronic cross sections in low-energy e+e- annihilations, which are accessible with data taken near the Upsilon(4S) via initial-state radiation. Our measurements allow significant improvements in the precision of the predicted value of the muon anomalous magnetic moment. These improvements are necessary for shedding light on the current ~3 sigma difference between the predicted and the experimental values. We have previously published results on a number of processes with two to six hadrons in the final state. Currently, the largest uncertainty on the calculation of the hadronic contribution in the energy region between 1 and 2 GeV stems from the e+ e- -> pi+ pi- pi0 pi0 cross section. A new precise measurement of this process is presented here, together with measurement of other low-multiplicity channels, such as e+ e- -> pi+ pi- eta.