The eigenfunctions (of the Laplacian) on various geometric spaces constitute a class of mathematical objects of fundamental importance. From the point of view of quantum mechanics, the eigenfunctions correspond to particles moving with a certain energy, which leads naturally to questions motivated by subfields of physics. For example, one also has the so-called sup-norm problem, which asks how high the peaks of an eigenfunctions can be. There is also the famous "Quantum Unique Ergodicity" problem for which Lindenstrauss won a Fields medal. In this talk, I will give a gentle introduction to some of these problems in a setting where number theory plays a key role. In the special case when the manifold is a surface of constant negative structure, and is constructed from "quaternion algebras", a famous result of Iwaniec and Sarnak improves upon the trivial bound for the sup-norm using number-theoretic techniques. I will explain this result, and then talk about recent progress on an analogous question where the underlying surface is itself allowed to vary (the level aspect). I will also explain the interesting connections between these questions and deep problems in number theory such as the subconvexity problem.