Spatio-temporal datasets are rapidly increasing in size. However they are often noisy and incomplete (nothing can be measured everywhere all the time), and so statistical inference is required to obtain complete maps of a spatio-temporal process of interest, together with proper uncertainty quantification. We focus here on (near-) real-time filtering inference in linear Gaussian state-space models, where the state at each time point is a spatial field evaluated on a very large spatial grid. For these models, exact inference using the Kalman filter is computationally infeasible. Instead, we propose a multi-resolution filter (MRF), which approximates the distribution of the spatial field using a large number of spatial basis functions at multiple resolutions, which are automatically determined to capture the spatial structure at all scales. The MRF is highly scalable in the size of the spatial grid, and it is well-suited for massively distributed computations. We also discuss parameter inference using a Rao-Blackwellized particle filter, in which the integrated likelihood is computed using the MRF.