In the world of space-bounded complexity, there is a strain of results showing that space can, somewhat paradoxically, be used for multiple purposes at once. Touchstone results include Barrington’s Theorem and the recent line of work on catalytic computing. We refer to such techniques, in contrast to the usual notion of reclaiming space, as reusing space.

In this survey we will dip our toes into the world of reusing space. We do so in part by studying techniques, viewed through the lens of a few highlight results, but our main focus will be the wide variety of open problems in the field.

In addition to the broader and more challenging questions, we aim to provide a number of questions that are fairly simple to state, have clear practical and theoretical implications, and, most importantly, that a newcomer with little background experience can still sit down and play with for a while.