What can symplectic topology tell us about algebraic varieties?

I will begin by briefly recalling the relationship between complex projective algebraic geometry and symplectic topology, which goes through Kaehler manifolds. I will then survey results from the end of the last century, largely due to Seidel and McDuff, about the symplectic topology of Hamiltonian fibrations over the 2-sphere, and their consequences for smooth projective maps over the projective line. Finally, I will indicate some recent advances in this area, including the use of methods of Floer homotopy theory to refine our knowledge about the topology of these spaces.