Various generalizations of primary ideals of a commutative ring R with  have been studied. For example, a proper ideal I of R is a 2-absorbing primary (resp., weakly 2-absorbing primary) if    with  (resp.,  implies  or  or  In this paper, we introduce the concept of almost2-absorbing primary ideal as a new generalization of primary ideals. A proper ideal I of R is called an almost 2-absorbing primary ideal of R whenever  and  then  or  or  A number of results concerning almost 2-absorbing primary ideals such as their properties and their relation among other primary ideals will be given. Also we will classify all rings for which every proper ideal is an almost 2-absorbing primary ideal.

primary ideal, prime ideal, 2-absorbing ideal, almost prime ideal, almost primary ideal, almost 2-absorbing ideal, almost 2-absorbing primary ideal.