We will review what is known in general about the relationship between the fluctuations of an observable classically and quantum mechanically, when the classical dynamics is chaotic. Specifiaccly we examine the question of the main term in Zelditch's variance sums and the physics conjectures of Feingold and Perez. We then specialize to the case of the mechanics of geodesic motion on a on a hyperbolic surface, (this being the only case where the question has now been settled) and explain the surprizing answer.