Hamiltonian Dynamical Systems Without Periodic Orbits

The present paper is a review of counterexamples to the ``Hamiltonian Seifert conjecture'' or, more generally, of examples of Hamiltonian systems having no periodic orbits on a compact energy level. We begin with the discussion of the ``classical'' and volume--preserving Seifert conjectures. Then a construction of counterexamples to the Hamiltonian Seifert conjecture in dimensions greater than or equal to six is outlined, mainly following the method introduced by the author of the paper. The Hamiltonian version of Seifert's theorem is also stated. A list of all known at this moment examples and constructions of Hamiltonian flows without periodic flows concludes the review.