Abstract: In recent years, one major focus of modeling spatial data has been to connect two contrasting approaches, namely, the Markov random field approach and the geostatistical approach. While the geostatistical approach allows flexible modeling of the spatial processes and can accommodate continuum spatial variation, it faces formidable computational burden for large spatial data. On the other hand, spatial Markov random fields facilitate fast statistical computations but they lack in flexibly accommodating continuum spatial variations. In this talk, I will discuss novel statistical models and methods which allow us to get the best of both worlds, i.e., they can accommodate continuum spatial variation at the same time allowing fast matrix-free statistical computations for large spatial data. Keeping various large spatial data in mind, I will first discuss graphical exploratory analysis with these models and then move on to formal likelihood based inference.