We present a Parametrized-Background Data-Weak (PBDW) formulation of the variational data assimilation (state estimation) problem for partial differential equations. The main contributions are a constrained optimization/weak framework informed by the notion of experimentally observable spaces; a priori and a posteriori error estimates for the field and associated linear-functional outputs; weak greedy construction of prior (background) spaces associated with an underlying and potentially high?dimensional parametric manifold; stability-informed choice of observation functionals and related sensor locations ; and finally, output prediction from the optimality saddle in O(M3) operations, where M is the number of experimental observations. We present results for several synthetic model problems to illustrate the elements of the methodology, confirm the numerical properties suggested by theory, and demonstrate the generality of the approach. To conclude, we consider a physical raised-box acoustic resonator problem: we integrate the PBDW methodology and a Robotic Observation Platform to achieve real-time in situ state estimation of the full pressure field. Work in collaboration with Yvon Maday, James Penn, Tommaso Taddei, and Masa Yano.