Abstract: My talk is about some recent results (joint work with C.Sogge, J. Toth and B. Shiffman) about asymptotics of eigenfunctions of the Laplacian on compact Riemannian manifolds. They are motivated by graphics of highly excited quantum states due to physicists working in quantum chaos and mescoscopic physics. The main question is how the the dynamics of the geodesic flow impacts on features of eigenfunctions. For instance, the exponentials on a flat torus have uniformly bounded sup norms while the zonal spherical harmonics on the standard sphere blow up quickly in the sup norm. We will prove that these examples illustrate what must happen in the extreme cases where sup norms are bounded or when they blow up at the maximal rate. We will also discuss zeros and critical points of eigenfunctions. The talk is meant for a general audience and includes computer graphics of eigenfunctions.