Under the influence of Kripke's acute analysis, there has been a growing trend of modern essentialism, or in other words, the assertion that there are 'essential' descriptors (rigid designators) that 'stick' with the thing being described in all possible worlds. Most influentially Kripke identified that any set of descriptions could not be definite in the way Russell imagined, since it was possible that a given X with descriptions Y and Z could have been existent in another world where Y and Z were not applicable to X.
Russell's descriptivist theory of names still holds a large degree of influence in analytical philosophy however. It is assumed especially in the way philosophers talk about existence. Existence is typically understood to be a general concept property which is instantiated in existent objects. This seems to largely hold only if we grant that existence cannot be attributed to singular things, which in turn typically only holds because singular things are held to be a collection of descriptions. If one were to rid of the last, grounding belief one would find new gates opened for conversation about what the nature of existence is.
In any case, it seems clear that Kripke's criticism has very serious and far-reaching consequences. As such, it is important to understand the responses that have been given to such criticism. What is the current debate being had about Russell's theory vs. Kripke's?