  We define a pseudometric on the set of all unbounded subsets of a metric
space. The Kolmogorov quotient of this pseudometric space is a complete metric
space. The definition of the pseudometric is guided by the principle that two
unbounded subsets have distance 0 whenever they stay sublinearly close. Based
on this pseudometric we introduce and study a general concept of boundaries of
metric spaces. Such a boundary is the closure of a subset in the Kolmogorov
quotient determined by an arbitrarily chosen family of unbounded subsets.
  Our interest lies in those boundaries which we get by choosing unbounded
cyclic sub-(semi)-groups of a finitely generated group (or more general of a
compactly generated, locally compact Hausdorff group). We show that these
boundaries are quasi-isometric invariants and determine them in the case of
nilpotent groups as a disjoint union of certain spheres (or projective spaces).
  In addition we apply this concept to vertex-transitive graphs with polynomial
growth and to random walks on nilpotent groups.
