  In this article we introduce and study uniform and non-uniform approximate
lattices in locally compact second countable (lcsc) groups. These are
approximate subgroups (in the sense of Tao) which simultaneously generalize
lattices in lcsc group and mathematical quasi-crystals (a.k.a. Meyer sets) in
lcsc abelian groups.
  We show that envelopes of strong approximate lattices are unimodular, and
that approximate lattices in nilpotent groups are uniform. We also establish
several results relating properties of approximate lattices and their
envelopes. For example, we prove a version of the Milnor-Schwarz lemma for
uniform approximate lattices in compactly-generated lcsc groups, which we then
use to relate metric amenability of uniform approximate lattices to amenability
of the envelope.
  Finally we extend a theorem of Kleiner and Leeb to show that the isometry
groups of higher rank symmetric spaces of non-compact type are QI rigid with
respect to finitely-generated approximate groups.
