  Suppose that $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}^d}$ is the solution to a
$d$-dimensional stochastic heat equation driven by a Gaussian noise that is
white in time and has a spatially homogeneous covariance that satisfies
Dalang's condition. The purpose of this paper is to establish quantitative
central limit theorems for spatial averages of the form $N^{-d} \int_{[0,N]^d}
g(u(t\,,x))\, \mathrm{d} x$, as $N\rightarrow\infty$, where $g$ is a
Lipschitz-continuous function or belongs to a class of locally-Lipschitz
functions, using a combination of the Malliavin calculus and Stein's method for
normal approximations. Our results include a central limit theorem for the {\it
Hopf-Cole} solution to KPZ equation. We also establish a functional central
limit theorem for these spatial averages.
