  Inspired by work surrounding Igusa's local zeta function, we introduce
topological representation zeta functions of unipotent algebraic groups over
number fields. These group-theoretic invariants capture common features of
established $p$-adic representation zeta functions associated with pro-$p$
groups derived from unipotent groups. We investigate fundamental properties of
the topological zeta functions considered here. We also develop a method for
computing them under non-degeneracy assumptions. As an application, among other
things, we obtain a complete classification of topological representation zeta
functions of unipotent algebraic groups of dimension at most six.
