We will discuss L^2 to L^q bounds for spectral projection operators on logarithmic intervals on hyperbolic manifolds. In dimension two, we will describe an application of these bounds to the L^q' to L^q mapping properties of the spectral measure on convex cocompact hyperbolic surfaces. If time permits, we will also discuss recent ongoing work on improved L^q estimates for arithmetic eigenfunctions. This is based on joint work with Jiaqi Hou, Chris Sogge, Zhexing Zhang and Zhongkai Tao.
