Comment: 60 pages, 4 tablesThe matrix exponential spatial models exhibit similarities to the conventional spatial autoregressive model in spatial econometrics but offer analytical, computational, and interpretive advantages. This paper provides a comprehensive review of the literature on the estimation, inference, and model selection approaches for the cross-sectional matrix exponential spatial models. We discuss summary measures for the marginal effects of regressors and detail the matrix-vector product method for efficient estimation. Our aim is not only to summarize the main findings from the spatial econometric literature but also to make them more accessible to applied researchers. Additionally, we contribute to the literature by introducing some new results. We propose an M-estimation approach for models with heteroskedastic error terms and demonstrate that the resulting M-estimator is consistent and has an asymptotic normal distribution. We also consider some new results for model selection exercises. In a Monte Carlo study, we examine the finite sample properties of various estimators from the literature alongside the M-estimator.