In this thesis, we study the two-point Weyl Law for the Laplace-Beltrami operator on a smooth, compact Riemannian manifold $M$ with no conjugate points. We find the asymptotic behavior of the Schwartz kernel, $\Pi_\lambda(x,y)$, of the projection operator from $L^2(M)$ onto the direct sum of eigenspaces with eigenvalue smaller than $\lambda^2$ as $\lambda \to\infty$. In the regime where $x,y$ are restricted to a sufficiently small compact neighborhood of the diagonal in $M\times M$, we obtain a uniform logarithmic improvement in the remainder of the asymptotic expansion for $\Pi_\lambda$ and its derivatives of all orders. This generalizes a result of B\'erard that established an on-diagonal estimate for $\Pi_\lambda(x,x)$ without derivatives. Furthermore, when $x,y$ avoid a compact neighborhood of the diagonal, we obtain the same logarithmic improvement in the standard upper bound for the derivatives of $\Pi_\lambda$. Our results imply that the rescaled covariance kernel of a monochromatic random wave locally converges in the $C^\infty$-topology to a universal scaling limit at an inverse logarithmic rate.