The stable Andrews-Curtis conjecture in combinatorial group theory is equivalent to the conjecture that every finite contractible 2-complex can be reduced to a single point by elementary expansions and collapses through complexes of dimension at most 3. In group-theoretic terms, this means that every presentation of the trivial group with equal numbers of generators and relators can be simplified to standard form by elementary moves corresponding to "addition of cells" or "handle-slides", together with "stabilization" moves that increase or decrease the number of generators (and relators). The role of the stabilization moves is unclear; conceivably any presentation that can be simplified using the full set of moves can also be simplified without stabilization. I will discuss several equivalent forms of the conjecture, and present (rather weak) evidence that presentations can be "improved" by stabilization.