On a Problem of Danzer In this talk, I will give a brief overview of the Hadwiger-Debrunner (p,q) theorem of Alon and Kleitman and related problems. The result of Alon and Kleitman states that if we have a set $S$ of convex sets in $\R^d$ such that out of every $p$ of the convex sets some $q$ have a common intersection, then there is a set of $t = HD(p,q)$ points that `hit' every convex set in $S$. Note that the number of points $t$ does not depend on the number of convex sets in $S$. This results holds for any $p \geq q \geq d+1$. Danzer studied the problem when $S$ is a set of balls in $\R^d$. I will present polynomial bounds on $HD(p,q)$ for this case, improving over a nearly $60$-year old exponential bound---roughly $O(2^d)$---of Danzer. This is joint work with Nabil Mustafa.