1 Introduction
1.1 Examples
1.2 Fractional Gaussian fields in one dimension
1.3 Interpretation as a long range GFF

2 Preliminaries
2.1 Tempered Distributions and Sobolev spaces
2.2 The Fractional Laplacian
2.3 White Noise

3 The FGF on 𝑅𝑑
3.1 Definition of 𝐹𝐺𝐹𝑠(𝑅𝑑)
3.2 The FGF covariance kernel

4 The FGF on a domain
4.1 The space 𝐻̇0𝑠(D)
4.2 The zero-boundary FGF in a domain
4.3 Covariance kernel for the FGF on the unit ball

5 Projections of the FGF

6 Fractional Brownian motion and the FGF

7 Restricting FGFs

8 Regularity of FGF(D)

9 The eigenfunction FGF

10 FGF local sets
10.1 FGF with Boundary Values
10.2 Local Sets of the FGF on a Bounded Domain
10.3 An example of a local set

11 Spherical decomposition
11.1 FGF spherical average processes
11.2 Background on spherical harmonic functions
11.3 Spherical decomposition of the FGF

12 The discrete fractional Gaussian field
12.1 Fractional gradient
12.2 The discrete fractional Gaussian field

13 Open questions
13.1 Questions on level sets
13.2 Other questions