Around Zariski cancellation problem

If the product of a complex affine algebraic variety X on an affine space is isomorphic to an affine space, is it true that X itself is isomorphic to an affine space? This question, posed in the early eighties of the last century, is still open. We will discuss several topics inspired by it, including the problem of classifying a class of affine algebraic groups that are naturally singled out in the study of algebraic subgroups of the Cremona groups and their conjugacy.