Observability is a well defined notion in the theory of controls for dynamical systems which, in a sense is dual to the idea of controllability. In particular observability plays an important role in data assimilation. After a short review of the basic ideas, I'll introduce a range of recent applications to partial differential equations arising in geophysics. I'll also present some recent work on image-based observers and other similar dynamically interesting quantities, which can also be used to infer the state of the system. The talk is aimed at a general scientific audience and will not assume more than a familiarity with differential equations.