How to (not) cancel varieties

Since the early 1970s, various authors have made steady progress toward resolutions of the algebraic cancellation problem, an easily-stated yet compellingly-elusive question whose difficulty emphasizes the limitations of our understanding of the geometry of objects as fundamental to algebraists as complex affine n-space. We survey these partial resolutions and examine the first known counterexample [5] in light of developments since the last prominent expository note [16] written on the subject.