is to provide a mathematically rigorous, unified, and accessible account of the $\operatorname{FGF}_s({\mathbb{R}}^d)$ processes, treating the full range of values $s\in {\mathbb{R}}$ and $d \in {\mathbb{N}}$. This paper is fundamentally a survey, but we also present several basic facts that we have not found articulated elsewhere in the literature. Many of these are generalizations of classical results that had previously only been formulated for specific $d$ and $s$ values.

We hope that this survey will increase the circulation of basic information about fractional Gaussian fields in the mathematical community. For example, the vocabulary and content of the following statements should arguably be well known to probabilists, but the authors were unaware of much of it until recently:

-   In dimension $3$, the Gaussian field with logarithmic correlations has been used as an approximate model for the gravitational potential of the early universe; its Laplacian is a $\operatorname{FGF}_{-1/2}({\mathbb{R}}^3)$ and has been used to model the perturbation from uniformity of the mass/energy density of the early universe[^3].

-   In dimension $4$, the so-called bi-Laplacian field has logarithmic correlations, and its Laplacian is white noise.

-   In any dimension, Lévy Brownian motion can be defined as a random continuous function whose restriction to 