  We discuss recent results of the replica approach to statistical mechanics of
a single classical particle placed in a random N(>>1)-dimensional Gaussian
landscape. The particular attention is paid to the case of landscapes with
logarithmically growing correlations and to its recent generalisations. Those
landscapes give rise to a rich multifractal spatial structure of the associated
Boltzmann-Gibbs measure. We also briefly mention related results on counting
stationary points of random Gaussian surfaces, as well as ongoing research on
statistical mechanics in a random landscape constructed locally by adding many
squared Gaussian-distributed terms.
