We study a class of conformally invariant operators $P$ and their related
conformally invariant curvatures $Q$ on even-dimensional Riemannian
manifolds. When the manifold is locally conformally flat(LCF) and compact
without boundary, $Q$-curvature is naturally related to the integrand in
the classical Gauss-Bonnet-Chern formula, i.e., the Pfaffian curvature
(CF. Branson, Branson-Gilkey-Pohjanpelto). For a class of even-dimensional
complete LCF manifolds with integrable $Q$-curvature, we establish a
Gauss-Bonnet-Chern inequality.  These are extensions of the classical
results of Cohn-Vossen and Huber in dimension two and those of
Chang-Qing-Yang in dimension four. As applications, finiteness theorems
for certain classes of complete LCF manifolds are also proven.