Many real-world datasets have spatial components that provide valuable information about characteristics of the data. In this dissertation, a novel framework for adaptive models that exploit spatial information in data is proposed. The proposed framework is mainly based on development and applications of Gaussian processes. First, a supervised learning method is proposed for the classification of hyperspectral data with spatially adaptive model parameters. The proposed algorithm models spatially varying means of each spectral band of a given class using a Gaussian process regression model. For a given location, the predictive distribution of a given class is modeled by a multivariate Gaussian distribution with spatially adjusted parameters obtained from the proposed algorithm. The Gaussian process model is generally regarded as a good tool for interpolation, but not for extrapolation. Moreover, the uncertainty of the predictive distribution increases as the distance from the training instances increases. To overcome this problem, a semi-supervised learning algorithm is presented for the classification of hyperspectral data with spatially adaptive model parameters. This algorithm fits the test data with a spatially adaptive mixture-of-Gaussians model, where the spatially varying parameters of each component are obtained by Gaussian process regressions with soft memberships using the mixture-of-Gaussian-processes model. The proposed semi-supervised algorithm assumes a transductive setting, where the unlabeled data is considered to be similar to the training data. This is not true in general, however, since one may not know how many classes may existin the unexplored regions. A spatially adaptive nonparametric Bayesian framework is therefore proposed by applying spatially adaptive mechanisms to the mixture model with infinitely many components. In this method, each component in the mixture has spatially adapted parameters estimated by Gaussian process regressions, and spatial correlations between indicator variables are also considered. In addition to land cover and land use classification applications based on hyperspectral imagery, the Gaussian process-based spatio-temporal model is also applied to predict ground-based aerosol optical depth measurements from satellite multispectral images, and to select the most informative ground-based sites by active learning. In this application, heterogeneous features with spatial and temporal information are incorporated together by employing a set of covariance functions, and it is shown that the spatio-temporal information exploited in this manner substantially improves the regression model. The conventional meaning of spatial information usually refers to actual spatio-temporal locations in the physical world. In the final chapter of this dissertation, the meaning of spatial information is generalized to the parametrized low-dimensional representation of data in feature space, and a corresponding spatial modeling technique is exploited to develop a nearest-manifold classification algorithm. / text