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This dissertation is comprised of an introductory chapter and three stand-alone chapters, tied together by a unifying theme: the statistical analysis of very large spatial and spatio-temporal datasets. These datasets now arise in many fields, but our focus here is on environmental remote-sensing data. Due tosparseness of daily datasets, there is a need to fill spatial gaps and to borrow strength from adjacent days. Nonetheless, many satellite instruments are capable of conducting on the order of 100,000 retrievals per day,which makes it computationally challenging to apply traditional spatial and spatio-temporalstatistical methods, even in supercomputing environments. In addition, the datasets are often observed on the entire globe. For such large domains, spatial stationarity assumptions are typically unrealistic. We address these challenges using dimension-reduction techniques based on a flexible spatial random effects (SRE) model, where dimension reduction is achieved by projecting the process onto a basis-function space of low dimension. The spatio-temporal random effects (STRE) model extends the SRE model to the spatio-temporal case by modeling the temporal evolution, on the reduced space, using a dynamical autoregressive model in time. Another focus of this work is the modeling of fine-scale variation. Such variability is typically not part of the reduced space spanned by the basis functions, and one needs to account for a component of variability at a fine scale. We address this issue throughout the dissertation with increasingly complex and realistic models for a component of fine-scale variation.After a general introductory chapter, the subsequent two chapters focus on estimation of the reduced-dimensional parameters in the STRE model from both an empirical-Bayes and a fully Bayesian perspective, respectively. In Chapter 2, we develop maximum likelihood estimation via an expectation-maximization (EM) algorithm, which offers stable computation of valid estimators and makes efficient use of spatial and temporal dependence in the data, assuming a multivariate Gaussian model. In Chapter 3, we develop a multiresolutional prior for the propagator matrix on the reduced-dimensional space that allows for unknown (random) sparsity and shrinkage, and we describe how sampling from the posterior distribution can be achieved in a feasible way, even if this matrix is very large.Finally, in Chapter 4, we return to the spatial-only case. We generalize the standard SRE model and provide informative prior distributions for the parameters of the generalized SRE model based on a nonstationary covariance model in physical space. We propose a comprehensive model that takes account of all scales of variation, in particular by allowing for the fine-scale-variation component to exhibit spatial dependence. We make inference on the number, locations, and shapes of the basis functions. Computational feasibility is maintained by assuming that the fine-spatial-scale covariance is compactly supported, resulting in a very sparse covariance matrix for the fine-scale-variation component.All methodological results are illustrated and compared using simulation studies and a dataset of global satellite measurements of CO2, which came from the Atmospheric InfraRed Sounder (AIRS) instrument on NASA's Aqua satellite.