In this thesis we study contact structures on smooth manifolds, i. e. completely nonintegrable subbundles of codimension one of the tangent bundle. We prove Gray's theorem, which states that the members of a smooth family of contact structures on a closed manifold are pairwise contactomorphic. Darboux's theorem, which we also prove, states that contact manifolds have no local invariants, apart from dimension. It is itself a special case of the isotropic neighbourhood theorem, which allows us to identify tubular neighbourhoods of diffeomorphic isotropic submanifolds in two different contact manifolds, provided they have isomorphic conformal symplectic normal bundles. This condition holds trivially for Legendrian submanifolds, which in turn allows a more detailed study of curves and knots in three-manifolds. We show that any continuous curve can be approximated with a Legendrian one in C0-topology. We prove that closed orientable three manifolds always admit contact structures. We define overtwisted contact structures on three-manifolds and study their basic properties.
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