  This paper concerns the asymptotic behavior of zeros and critical points for
monochromatic random waves $\phi_\lambda$ of frequency $\lambda$ on a compact,
smooth, Riemannian manifold $(M,g)$ as $\lambda \rightarrow \infty$. We prove
that the measure of integration over the zero set of $\phi_\lambda$ restricted
to balls of radius $\approx \lambda^{-1}$ converges in distribution to the
measure of integration over the zero set of a frequency $1$ random wave on
$\mathbb R^n$, where $n$ is the dimension of $M$. We also prove convergence of
finite moments for the counting measure of the critical points of
{\phi}{\lambda}, again restricted to balls of radius $\approx \lambda^{-1}$, to
the corresponding moments for frequency $1$ random waves. We then patch
together these local results to obtain new global variance estimates on the
volume of the zero set and numbers of critical points of $\phi_\lambda$ on all
of $M.$ Our local results hold under conditions about the structure of
geodesics on $M$ that are generic in the space of all metrics on $M$, while our
global results hold whenever $(M,g)$ has no conjugate points (e.g is negatively
curved).
