The topic of forgetting, which loosely speaking means losing, removing, or even hiding some variables, propositions, or formulas, has been extensively studied in the field of knowledge representation and reasoning for many major formalisms. In this article, we convey this topic to the highly active field of abstract argumentation. We provide an in-depth analysis of desirable syntactical and/or semantical properties of possible forgetting operators. In doing so, we included well-known logic programming conditions, such as strong persistence or strong invariance. Further, we argue that although abstract argumentation and logic programming are closely related, it is not possible to reduce forgetting in abstract argumentation to forgetting in logic programming in a straightforward manner. The analysis of desiderata, adapted to the specifics of abstract argumentation, includes implications among them, individual and collective satisfiability, and identifying inherent limits for a set of prominent semantics. Finally, we conduct a case study on stable semantics incorporating concrete forgetting operators.