Abstract: We generate new standard solar models using newly analyzed nuclear fusion cross sections and present results for helioseismic quantities and solar neutrino fluxes. The status of the solar abundance problem is discussed. We investigate whether nonstandard solar models with accretion from the protoplanetary disk might alleviate this problem. We examine a broad range of models, analyzing metal-enriched and metal-depleted accretion and three scenarios for the timing of accretion. Only partial solutions are found. Formetal-rich accreted material (Z(ac) greater than or similar to 0.018) there exist combinations of accreted mass and metallicity that bring the depth of the convective zone into agreement with the helioseismic value. For the surface helium abundance, the helioseismic value is reproduced if metal-poor or metal-free accretion is assumed (Z(ac) less than or similar to 0.09). In both cases a few percent of the solar mass must be accreted. Precise values depend on when accretion takes place. We do not find a simultaneous solution to both problems but speculate that changing the hydrogen-to-helium mass ratio in the accreted material may lead to more satisfactory solutions. We also show that, with current data, solar neutrinos are already a very competitive source of information about the solar core and can help constraining possible accretion histories. Even without helioseismic constraints, solar neutrinos rule out the possibility that more than 0.02 M(circle dot) from the protoplanetary disk were accreted after the Sun settled on the main sequence. Finally, we discuss how measurements of neutrinos from the CN cycle could shed light on the interaction between the early Sun and its protoplanetary disk.