  We survey the properties of the log-correlated Gaussian field (LGF), which is
a centered Gaussian random distribution (generalized function) $h$ on $\mathbb
R^d$, defined up to a global additive constant. Its law is determined by the
covariance formula $$\mathrm{Cov}\bigl[ (h, \phi_1), (h, \phi_2) \bigr] =
\int_{\mathbb R^d \times \mathbb R^d} -\log|y-z| \phi_1(y) \phi_2(z)dydz$$
which holds for mean-zero test functions $\phi_1, \phi_2$. The LGF belongs to
the larger family of fractional Gaussian fields obtained by applying fractional
powers of the Laplacian to a white noise $W$ on $\mathbb R^d$. It takes the
form $h = (-\Delta)^{-d/4} W$. By comparison, the Gaussian free field (GFF)
takes the form $(-\Delta)^{-1/2} W$ in any dimension. The LGFs with $d \in
\{2,1\}$ coincide with the 2D GFF and its restriction to a line. These objects
arise in the study of conformal field theory and SLE, random surfaces, random
matrices, Liouville quantum gravity, and (when $d=1$) finance. Higher
dimensional LGFs appear in models of turbulence and early-universe cosmology.
LGFs are closely related to cascade models and Gaussian branching random walks.
We review LGF approximation schemes, restriction properties, Markov properties,
conformal symmetries, and multiplicative chaos applications.
