A topological fibration of CW complexes has a continuous section if the base is r-dimensional and the fiber is (r-1)-connected, e.g., path connected when r=1 and simply connected when r=2. I will explain analogues in algebraic geometry for r=1, 2, as well as some consequences: an algebraic fibration of complex projective varieties has a rational section if the base is a curve, resp. a surface, and the generic fiber is rationally connected, resp. rationally simply connected. This project is joint work with Graber and Harris (for r=1) and with de Jong and Xuhua He (for r=2).