A 1965 problem due to Danzer asks whether there exists a set with finite density in Euclidean space (i.e. « not containing too many points ») intersecting any convex body of volume one. A suitable weakening of the volume constraint leads one to the (much more recent) problem of constructing dense forests. Progress towards these problems have so far involved a very wide range of areas in mathematics (including number theory, ergodic theory and dynamical systems). After surveying some of the known results related to the Danzer Problem and to the construction of dense forests, the talk will present some new constructions.