We discuss recent results of the replica approach to statistical mechanics of a single classical particle placed in a random N (\gg 1)-dimensional Gaussian landscape. The particular attention is paid to the case of landscapes with logarithmically growing correlations and to its recent generalizations. 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.

PACS numbers: 05.40.--a, 75.10.Nr