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ABSTRACT: In his pioneering work, using pseudoholomorphic curves in the symplectisations, Hofer studied Hamiltonian flows restricted to an energy level which is of contact type. He then proved many cases of Weinstein's conjecture on the existence of a periodic orbit in dimension three. The theory further developed by Hofer, Wysocki and Zehnder, and since then, pseudoholomorphic curves in symplectisations have been used as a powerful tool to study Hamiltonian systems with two degrees of freedom. In particular, they introduced a notion of a transverse foliation, which is a singular foliation of an energy level, where the singular set consists of finitely many periodic orbits, called bindings, and the regular leaves are punctured Riemann surfaces transverse to the flow and asymptotic to the bindings at the punctures. This allows one to reduce the Hamiltonian dynamics on the three-dimensional energy level to the study of an area-preserving surface map. In this lecture series, I will provide a gentle introduction on the theory of Hofer, Wysocki and Zehnder and explain how to construct transverse foliations in various concrete problems such as the restricted three-body problem. This is based on joint work with Naiara de Paulo, Umberto Hryniewicz, Pedro Salomão and Alexsandro Schneider.