Our goal is to relate the observation (or control) of the wave equation on observation domains which evolve in time with some dynamical properties of the geodesic flow. In comparison to the case of static domains of observation, we show that the observability of the wave equation in any dimension of space can be improved by allowing the domain of observation to move. We first prove that, for any domain Ω satisfying a geodesic recurrence condition (GRC), it is possible to observe the wave equation in infinite time on a ball of radius ε moving in Ω at finite speed v, where ε > 0 and v > 0 can be taken arbitrarily small, whereas the wave equation in Ω may not be observable on any static ball of radius ε. We comment on the recurrence condition: we give examples of Riemannian manifolds (Ω, g) for which (GRC) is satisfied, and, using a construction inspired by the Birkhoff-Smale homoclinic theorem, we show that there exist Riemannian manifolds (Ω, g) for which (GRC) is not satisfied. Then we prove that on the 2-dimensional torus and on Zoll manifolds, it is possible to observe the wave equation in finite time with moving balls. Finally, we establish a result of spectral observability (or of concentration of eigenfunctions) on time-dependent domains.