For over forty years, the Fay-Herriot model has been extensively used by National Statistical Offices around the world to produce reliable small area statistics. This model develops prediction of small area means of a continuous outcome of interest based on a linear regression on suitable auxiliary variables. Model errors, also known as small area effects, of the Fay-Herriot model are treated as independent and normally distributed zero-mean random variables with an unknown variance. Often population means of geographically contiguous small areas display a spatial pattern. The independence assumption for the random effects may not hold when effective auxiliary variables are unavailable. Lack of suitable covariates to account for the variation of the geographic domain means results in a spatial pattern among the random effects. We consider several spatial random-effects models, including the popular conditional autoregressive and simultaneous autoregressive models as alternatives to the Fay-Herriot model. We carry out a Bayesian analysis of these models based on a class of popular noninformative improper prior densities for the model parameters. We assess the effectiveness of these spatial models based on a simulation study and a real application. We consider the prediction of statewide four-person family median incomes for the U.S. states based on the 1990 Current Population Survey and the 1980 Census. This application and simulation study show considerably superior performance of some of the spatial models over the regular Fay-Herriot model when good covariates remain unavailable. In some applications, some small areas are created after the completion of a survey that does not provide any direct estimates of the late-breaking unsampled small areas. Proposed spatial models generate better predictions of unsampled small area means by borrowing from neighboring residuals than the synthetic regression means that result from the regular independent random effects Fay-Herriot model. For all the spatial Bayesian models considered, their posterior distributions based on a useful class of improper prior densities on model parameters, even in the absence of data from some small areas, are shown to be proper under some mild conditions.