  The main result of this paper is a bound on the distance between the
distribution of an eigenfunction of the Laplacian on a compact Riemannian
manifold and the Gaussian distribution. If $X$ is a random point on a manifold
$M$ and $f$ is an eigenfunction of the Laplacian with $L^2$-norm one and
eigenvalue $-\mu$, then $$d_{TV}(f(X),Z)\le\frac{2}{\mu}\E\big|\|\nabla
f(X)\|^2-\E\|\nabla f(X) \|^2\big|.$$ This result is applied to construct
specific examples of spherical harmonics of arbitrary (odd) degree which are
close to Gaussian in distribution. A second application is given to random
linear combinations of eigenfunctions on flat tori.
