Recent years have seen a marked increase in the application of spatial models in economics and the social sciences, in general. However, the development of a general asymptotic estimation and inference theory for spatial estimators has been hampered by a lack of central limit theorems (CLTs), uniform laws of large numbers (ULLNs) and pointwise laws of large number (LLN) for random fields under the assumptions relevant to economic applications. These limit theorems are the basic building blocks for the asymptotic theory of M-estimators, including maximum likelihood and generalized method of moments estimators. The dissertation derives new CLTs, ULLNs and LLNs for weakly dependent random fields that are applicable to a broad range of data processes in economics and other fields. Relative to the existing literature, the contribution of the dissertation is threefold. First, the proposed limit theorems accommodate nonstationary random fields with asymptotically unbounded or trending moments. Second, they cover a larger class of weakly dependent spatial processes than mixing random fields. Third, they allow for arrays of fields located on unevenly spaced lattices, and place minimal restrictions on the configuration and growth behavior of index sets. Each of the theorems is provided with weak yet primitive sufficient conditions.