Let \$V\$ be a projective variety over a number field, and let \$Z\$ be a subset of \$V\$ of codimension at least two. In 2001, Hassett and Tschinkel asked whether or not it was true that the rational points of \$V\$ are potentially Zariski dense if and only if the \$Z\$-integral points of \$V\$ are potentially Zariski dense. Like the answers to every other question in arithmetic geometry, the answer is “sometimes yes, but most of the time we have no idea”. In this talk, I’ll give an overview of what’s known and what’s tantalizingly unknown. And despite the ominous word “puncturing”, no arithmetic was harmed in the making of this talk.