  We consider the boundary-path groupoids of topological higher-rank graphs. We
show that the all such groupoids are topologically amenable. We deduce that the
C*-algebras of topological higher-rank graphs are nuclear and prove versions of
the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness
theorem. We then provide a necessary and sufficient condition for simplicity of
a topological higher-rank graph C*-algebra, and a condition under which it is
also purely infinite.
