Quantum Ergodicity results have their origin in mathematical physics. The Quantum Unique Ergodicity of Rudnick and Sarnak is now resolved for the case of arithmetic Riemann surfaces by Lindenstrauss and Soundararajan. Prime geodesic theorems describe the asymptotic behaviour of primitive closed geodesics on hyperbolic manifolds and can be viewed as geometric analogues of the Prime number theorem. In this talk I will describe some of our recent work on these two problems for arithmetic 3-manifolds. Using triple product formulas and the Kuznetsov trace formula, the study of these two problems can be reduced to subconvexity estimates for related L-functions.