This talk provides an accessible overview of recent developments in the theory of scalar and geometric wave equations with random initial data. This area of research lies at the intersection of differential geometry, partial differential equations, and probability theory, and therefore connects to many beautiful topics in mathematics. In the first part of this talk, we review Hamiltonian equations on both Euclidean space and manifolds. Furthermore, we explain how basic results for Hamiltonian equations serve as a motivation for intricate questions on nonlinear wave equations. In the second and third parts of this talk, we state separate theorems for a scalar and a geometric wave equation, respectively. Without discussing any technical aspects, we will try highlight the interplay of different techniques needed for their proofs.