The question we are interested in is the following: Given an Artinian local ring, how ‘close’ can we get to it by an Artinian Gorenstein local ring. In this talk I will make the notion of being ‘close’ precise numerically. We will exhibit some natural bounds on this number and discuss some (old and new) results. In particular, if R is a quotient of a power series ring over a field of characteristic zero by a power of the maximal ideal, we will see how one can compute this number.