Over the last decade, convolution-based models for spatial data have increased in popularity as a result of their flexibility in modeling spatial dependence and their ability to accommodate large datasets. The modeling flexibility is due to the framework’s moving-average construction that guarantees a valid (i.e., non-negative definite) spatial covariance function. This constructive approach to spatial modeling has been used (1) to provide an alternative to the standard classes of parametric variogram/covariance functions commonly used in geostatistics; (2) to specify Gaussian-process models with nonstationary and anisotropic covariance functions; and (3) to create non-Gaussian classes of models for spatial data. Beyond the flexible nature of convolution-based models, computational challenges associated with modeling large datasets can be alleviated in part through dimension reduction, where the dimension of the convolved process is less than the dimension of the spatial data. In this paper, we review various types of convolution-based models for spatial data and point out directions for future research.