  A 1965 problem due to Danzer asks whether there exists a set with finite
density in Euclidean space intersecting any convex body of volume one. A
suitable weakening of the volume constraint leads to the (much more recent)
problem of constructing \emph{dense forests}. These are discrete point sets
getting uniformly close to long enough line segments.
  Progress towards these problems have so far involved a wide range of ideas
surrounding areas as varied as combinatorial and computation geometry, convex
geometry, Diophantine approximation, discrepancy theory, the theory of
dynamical systems, the theory of exponential sums, Fourier analysis,
homogeneous dynamics, the mathematical theory of quasicrystals and probability
theory.
  The goal of this paper is to survey the known results related to the Danzer
Problem and to the construction of dense forests, to generalise some of them
and to state a number of open problems to make further progress towards a
solution to this longstanding question.
