Cyber-aggression, cyberbullying, and cyber-grooming are distinctive and similar phenomena that represent the objectionable content appearing on online social media. Timely detection of the objectionable content is very important for its prevention and reduction. This article explores and spotlights diversity of definitions of cyber-aggression, cyberbulling, and cyber-grooming; analyzes current categorization systems and taxonomies; identifies the targets, target categories, and subcategories of the subjects of the objectionable content research; analyzes the ambiguity of the linguistic terms in the domain; reviews present databases gathered for researching the field; explores types of features used for modeling systems for automatic detection; and examines methods for automatic detection and/or prediction of the objectionable content. The results point to directions of system development for tracing transformations of objectionable content over time on different online social platforms.