  This set of lectures aims to give an overview of Donaldson's theory of linear
systems on symplectic manifolds and the algebraic and geometric invariants to
which they give rise. After collecting some of the relevant background, we
discuss topological, algebraic and symplectic viewpoints on Lefschetz pencils
and branched covers of the projective plane. The later lectures discuss
invariants obtained by combining this theory with pseudo-holomorphic curve
methods.
