Abstract: Eigenfunctions of the Laplace operator on a compact Riemannian manifold occur as shapes of 'pure' vibrations of a drum, or as pure states of a quantum mechanical system. Except for a few simple cases, they cannot be calculated explicitly. In this talk we will focus on the question in the title, where 'big' refers to the ratio of the maximum and the mean square of an eigenfunction, and the issue is what happens as the eigenvalue gets large. We will describe some new and some old results and conjectures; in particular, it will be seen that the answers depend on the dynamics of the geodesic flow on the manifold, as is suggested by physical considerations.