I will first present the class of almost Gorenstein rings, focusing on the one-dimensional case. This notion was introduced by Barucci and Froberg for algebroid curves and it has been recently generalized (by Goto and others) and widely studied for more general one-dimensional rings and, successively, in any dimension. 

In the one-dimensional case, other possible generalizations of Gorenstein and almost Gorenstein has been proposed; these classes are close to Gorenstein and almost Gorenstein rings under different aspects. I will describe one of this new classes, whose definition is motivated by the relationships between the properties of a local one-dimensional ring (R,m) and those of the R-algebra m:m.