Non-Gaussian spatial data arise in a number of disciplines. Examples include spatial data on disease incidences (counts), and satellite images of ice sheets (presence-absence). Spatial generalized linear mixed models (SGLMMs), which build on latent Gaussian processes or Markov random fields, are convenient and flexible models for such data and are used widely in mainstream statistics and other disciplines. For high-dimensional data, SGLMMs present significant computational challenges due to the large number of dependent spatial random effects. I will discuss projection-based approaches that reparameterize and reduce the number of random effects in SGLMMs, resulting in a dramatic reduction in computational costs for Bayesian and maximum likelihood inference. Our approach also addresses spatial confounding issues. This talk is based on joint work with Yawen Guan (SAMSI) and John Hughes (U of Colorado-Denver).