The author gives a definition of a Hamiltonian system with input and output. Let \((M,\omega)\) (state space) and \((W,\omega^e)\) (external input-output space) be symplectic manifolds, \({\dot \omega}\) be a symplectic form generated on \(TM\), \(\Omega =\pi^*_1{\dot \omega}- \pi^*_2\omega^e\) is a symplectic form on \(TM\times W\), where \(\pi_1\), \(\pi_2\) are natural projections. The external influence is given by a bundle \(\pi\colon B\to M\) (inputs) and a mapping \(f\colon B\to TM\times W\). The system is called full (degenerate) Hamiltonian if \(f(B)\) is a Lagrangian manifold (a submanifold of \(f'(B')\) for another full system) in \((TM\times W,\Omega)\). The interconnections between these systems and Lagrange systems with external forces are investigated, the relationship with network theory is shown, the applications to the study of symmetries and to realization theory for Hamiltonian systems are given.