Determining the Equation of State (EOS) of dense neutron-rich nuclear matter is a shared goal of both nuclear physics and astrophysics. Except possible phase transitions, the density dependence of nuclear symmetry Esym(ρ) is the most uncertain part of the EOS of neutron-rich nucleonic matter especially at supra-saturation densities. Much progresses have been made in recent years in predicting the symmetry energy and understanding why it is still very uncertain using various microscopic nuclear many-body theories and phenomenological models. Simultaneously, significant progresses have also been made in probing the symmetry energy in both terrestrial nuclear laboratories and astrophysical observatories. In light of the GW170817 event as well as ongoing or planned nuclear experiments and astrophysical observations probing the EOS of dense neutron-rich matter, we review recent progresses and identify new challenges to the best knowledge we have on several selected topics critical for understanding astrophysical effects of the nuclear symmetry energy.PACS. 2 6.60.Kp Contents B.A Li, P.G. Krastev, D.H. Wen and N.B. Zhang: Astrophysical Effects of Nuclear Symmetry Energy 5.2.2 Predicted correlation strength between the radii of neutron stars and the symmetry energy from low to high densities 32 5.2.3 Predicted effects of the symmetry energy on the tidal deformability of neutron stars 33 5.3 Post-GW170817 analyses of tidal deformability and radii of neutron stars as well as constraints on the nuclear EOS and symmetry energy . . . 34 5.3.