Colloquium: Status of α-particle condensate structure of the Hoyle state

The present understanding of the structure of the Hoyle state in C-12 is reviewed. Most of the theoretical approaches to the Hoyle state are shortly summarized. The corresponding results are analyzed with respect to whether they give evidence to the alpha-particle condensation structure of the Hoyle state (and other Hoyle-like states in heavier self-conjugate nuclei) or not.