  We investigate properties of closed approximate subgroups of locally compact
groups, with a particular interest for approximate lattices i.e. those
approximate subgroups that are discrete and have finite co-volume.
  We prove an approximate subgroup version of Cartan's closed-subgroup theorem
and study some applications. We give a structure theorem for closed approximate
subgroups of amenable groups in the spirit of the Breuillard--Green--Tao
theorem. We then prove two results concerning approximate lattices: we extend
to amenable groups a structure theorem for mathematical quasi-crystals due to
Meyer; we prove results concerning intersections of radicals of Lie groups and
discrete approximate subgroups generalising theorems due to Auslander,
Bieberbach and Mostow. As an underlying theme, we exploit the notion of good
models of approximate subgroups that stems from the work of Hrushovski, and
Breuillard, Green and Tao. We show how one can draw information about a given
approximate subgroup from a good model, when it exists.
