  We introduce geometric consideration into the theory of formal languages. We
aim to shed light on our understanding of global patterns that occur on
infinite strings. We utilise methods of geometric group theory. Our emphasis is
on large scale geometries. Two infinite strings have the same large scale
geometry if there are colour preserving bi-Lipschitz maps with distortions
between the strings. Call these maps quasi-isometries. Introduction of large
scale geometries poses several questions. The first question asks to study the
partial order induced by quasi-isometries. This partial order compares large
scale geometries; as such it presents an algebraic tool for classification of
global patterns. We prove there is a greatest large scale geometry and
infinitely many minimal large scale geometries. The second question is related
to understanding the quasi-isometric maps on various classes of strings. The
third question investigates the sets of large scale geometries of strings
accepted by computational models, e.g. B\"uchi automata. We provide an
algorithm that describes large scale geometries of strings accepted by B\"uchi
automata. This links large scale geometries with automata theory. The fourth
question studies the complexity of the quasi-isometry problem. We show the
problem is $\Sigma_3^0$-complete thus providing a bridge with computability
theory. Finally, the fifth question asks to build algebraic structures that are
invariants of large scale geometries. We invoke asymptotic cones, a key concept
in geometric group theory, defined via model-theoretic notion of ultra-product.
Partly, we study asymptotic cones of algorithmically random strings thus
connecting the topic with algorithmic randomness.
