Abstract: Analytic number theory began with studying the distribution of prime numbers, but it has evolved and grown into a rich subject lying at the intersection of analysis, algebra, combinatorics, and representation theory. Part of its allure lies in its abundance of problems which are tantalizingly easy to state which quickly lead to deep mathematics, much of which revolves around the study of L-functions. These extensions of the elusive Riemann zeta function $\zeta(s)$ are generating functions with multiplicative structure arising from either arithmetic-geometric objects (like number fields or elliptic curves) or representation-theoretic objects (automorphic forms). Many equidistribution problems in number theory rely on one's ability to accurately bound the size of L-functions; optimal bounds arise from the (unproven!) Riemann Hypothesis for $\zeta(s)$ and its extensions to other L-functions. I will discuss some motivating problems along with recent work (joint with Kannan Soundararajan) which produces new bounds for L-functions by proving a suitable "statistical approximation" to the (extended) Riemann Hypothesis.