We introduce a novel class of Gaussian random fields (GRFs), called generalized integrated Brownian fields (GIBFs), focusing on the use of GIBFs for Gaussian process regression in deterministic and stochastic simulation metamodeling. We build GIBFs from the well-known Brownian motion and discuss several of their properties, including differentiability that cart differ in each coordinate, no mean reversion, and the Markov property. We explain why we desire to use GRFs with these properties and provide formal definitions of mean reversion and the Markov property for real-valued, differentiable random fields. We show how to use GIBFs with stochastic kriging, covering trend modeling and parameter fitting, discuss their approximation capability, and show that the resulting metamodel also has differentiability that can differ in each coordinate. Last, we use several examples to demonstrate superior prediction capability as compared with the GRFs corresponding to the Gaussian and Matern covariance functions.