E. Giroux has showed that the study of contact structures is equivalent to that of open book decompositions on 3-manifolds. In this thesis, we discuss two applications of this correspondence in the low-dimensional topology.

In the first part, we show that every compact smooth 4-manifold X has a structure of a Broken Lefschetz Fibration. Furthermore, if [special characters omitted](X) > 0 then it also has a Broken Lefschetz Pencil structure with nonempty base locus. This improves a theorem of Auroux, Donaldson and Katzarkov, and our construction uses only the 4-dimensional handlebody theory.

In the second part, we study some invariants of contact structures that arise from the Giroux correspondence. We show that the Ozsváth-Szabó contact invariant c⁺(ξ) ∈ HF⁺ (−Y) of a contact 3-manifold (Y, ξ) can be calculated combinatorially if Y is the boundary of a certain type of plumbing X, and ξ is induced by a Stein structure on X. Our technique uses an algorithm of Ozsváth and Szabó to determine the Heegaard-Floer homology of such 3-manifolds. We discuss two important applications of this technique in contact topology. First, we show that it simplifies the calculation of the Ozsváth-Stipsicz-Szabó obstruction to admitting a planar open book. Then we define a numerical invariant of contact manifolds that respects a partial ordering induced by Stein cobordisms. We do a sample calculation showing that the invariant can get infinitely many distinct values.