  Hamiltonian systems, when coupled {\it via} time--delayed interactions, do
not remain conservative. In the uncoupled system, the motion can typically be
periodic, quasiperiodic or chaotic. This changes drastically when delay
coupling is introduced since now attractors can be created in the phase space.
In particular for sufficiently strong coupling there can be amplitude death
(AD), namely the stabilization of point attractors and the cessation of
oscillatory motion. The approach to the state of AD or oscillation death is
also accompanied by a phase--flip in the transient dynamics. A discussion and
analysis of the phenomenology is made through an application to the specific
cases of harmonic as well as anharmoniccoupled oscillators, in particular the
H\'enon-Heiles system.
