81 pages, 2 tables, 33 figures.-- Available online Dec 15, 2007.-- Review paper.The study of nuclear decay modes reveals a large panel of nuclear structure phenomena and allows us to investigate the behaviour of the atomic nucleus with an extreme imbalance of the number of neutrons and protons with respect to stable nuclei. In the present paper, we review certain aspects of β-delayed decay modes, of one- and two-proton radioactivity, and of the experimental techniques which allowed us to get deep insights into the organisation of the atomic nucleus. In most cases, the study of these decay modes is the only means to obtain the information searched for. The investigation of nuclear decay modes is shown to be a powerful tool to study the most proton-rich nuclei and their nuclear structure.The present work took advantage of many discussions we had with many colleagues all over the world who provided us with new insights, explanations, not yet published data, allowed us to include figures from their papers in this review and more. Their help and generosity are gratefully acknowledged here.Peer reviewed