Stationary Gaussian processes (GPs) have long been a staple as priors for spatial fields to be inferred in a wide variety of inverse problems. GPs are proper distributions, defined by their mean and covariance functions. GPs have also proven remarkably extendable, leading to innovations such as deep GPs and Vecchia-based approximations for large systems. While the standard, stationary GP has an established track record, prominent researchers from the past (e.g. Matheron in the '70s, Besag in the '90s) have advocated intrinsic GPs in their stead. These intrinsic formulations are improper, and add some advantages, along with theoretical and computational hurdles. This talk will (re)introduce intrinsic GPs and explore how they might be used in Bayesian inverse problems and how resulting posteriors compare to stationary GP-based formulations. This is joint work with a number of colleagues at Virginia Tech.