Knowledge representation and reasoning have a long tradition in the field of artificial intelligence. More recently, the aspect of forgetting, too, has gained increasing attention. Humans have developed extremely effective ways of forgetting e.g. outdated or currently irrelevant information, freeing them to process ever-increasing amounts of data. The purpose of this paper is to present abstract formalizations of forgetting operations in a generic axiomatic style. By illustrating, elaborating, and identifying different kinds and aspects of forgetting from a common-sense perspective, our work may be used to further develop a general view on forgetting in AI and to initiate and enhance the interaction and exchange among research lines dealing with forgetting, both, but not limited to, in computer science and in cognitive psychology.