  This paper concerns the asymptotic behavior of a random variable $W_\lambda$
resulting from the summation of the functionals of a Gibbsian spatial point
process over windows $Q_\lambda \uparrow R^d$. We establish conditions ensuring
that $W_\lambda$ has volume order fluctuations, that is they coincide with the
fluctuations of functionals of Poisson spatial point processes. We combine this
result with Stein's method to deduce rates of normal approximation for
$W_\lambda$, as $\lambda\to\infty$. Our general results establish variance
asymptotics and central limit theorems for statistics of random geometric and
related Euclidean graphs on Gibbsian input. We also establish similar limit
theory for claim sizes of insurance models with Gibbsian input, the number of
maximal points of a Gibbsian sample, and the size of spatial birth-growth
models with Gibbsian input.
