  We initiate a study of asymptotic dimension for locally compact groups. This
notion extends the existing invariant for discrete groups and is shown to be
finite for a large class of residually compact groups. Along the way, the
notion of Hirsch length is extended to topological groups and classical results
of Hirsch and Malcev are extended using a topological version of the
Poincar\'{e} lemma. We show that polycyclic-by-compact groups and compactly
generated, topologically virtually nilpotent groups are residually compact, and
that compactly generated nilpotent groups are polycyclic-by-compact. We prove
that for compactly generated, solvable-by-compact groups the asymptotic
dimension is majorized by the Hirsch length, and equality holds for
polycyclic-by-compact groups. We extend the class of elementary amenable groups
beyond the discrete case and show that topologically elementary amenable groups
with finite Hirsch length have finite asymptotic dimension. We prove that a
topologically elementary amenable group of finite Hirsch length with no
nontrivial locally elliptic normal closed subgroup is solvable-by-compact.
Finally, we show that a totally disconnected, locally compact, second countable
[SIN]-group has finite asymptotic dimension, if all of its discrete quotients
are so.
