Abstract: Is there a discrete set Y in R^d, with growth rate O(T^d), that intersects every convex set of volume 1? This question is due to Danzer, and it is open since the sixties. I'll present our progress regarding this question, which includes both negative and positive results. On one hand we are able to rule out the candidacy of certain constructions of uniformly discrete sets that arise in natural mathematical constructions, such as sets that correspond to substitution tilings, and cut-and-project sets. On the other hand we construct a Danzer set of growth rate O(T^d(logT)) for every d, improving the previous result from 71' that gave a growth rate of O(T^d(logT)^{d-1}). This is a joint work with Barak Weiss.