  The paper briefly presents history, status, and plans of the search for the
critical structures - the onset of fireball, the onset of deconfinement, and
the deconfinement critical point - in high energy nucleus-nucleus collisions.
First, the basic ideas are introduced, the history of the observation of
strongly interacting matter in heavy ion collisions is reviewed, and the path
towards the quark-gluon plasma discovery is sketched. Then the status of the
search for critical structures is discussed - the discovery of the onset of
deconfinement, indications for the onset of fireball, and still inconclusive
results concerning the deconfinement critical point. Finally, an attempt to
formulate priorities for future measurements - charm quarks vs the onset of
deconfinement and detailed study of the onset of fireball - closes the paper.
