This paper draws its motivation from applications in geophysics, agricultural, and environmental sciences where empirical evidence of slow decay of correlations have been found for data observed on a regular lattice. Spatial ARFIMA models represent a widely used class of spatial models for analyzing such data. Here, we consider their generalization to conditional autoregressive fractional integrated moving average (CARFIMA) models, a larger class of long memory models which allows a wider range of correlation behavior. For this class we provide detailed descriptions of important representative models, make the necessary comparison with some other existing models, and discuss some important inferential and computational issues on estimation, simulation and long memory process approximation. Results from model fit comparison and predictive performance of CARFIMA models are also discussed through a statistical analysis of satellite land surface temperature data.