Basic information about the theory of mono- and multifractal sets is presented. Geometric and thermodynamic descriptions are developed. The geometric picture is presented on the basis of the simplest examples of the Koch and Cantor fractal sets. An ultrametric space, representing the metric of a fractal set, is introduced on the basis of Cayley's hierarchical tree. The spectral characteristics of a multifractal formation are described. Attention is focused mainly on the application of the fractal concept for a thermodynamic system with partial memory loss, turbulent fluid flow, hierarchically coordinated set of statistical ensembles, Anderson's transition, and incommensurable and quasicrystalline structures.