A review is given of low-temperature studies of the properties of the two-dimensional electron gas in inversion layers on high-index Si surfaces. The long crystallographic periods associated with such surfaces produce the superlattice effect in the spectrum of the quasi-two-dimensional electron gas. The kinetic coefficients are found to exhibit singularities when the Fermi level and minigaps cross. This was established by measuring the static and the high-frequency conductivities, the Shubnikov--de Haas oscillations, the photoresistivity, the emission of thermal electrons, and the cyclotron resonance in $n$-type inversion layers near (001) for electron concentrations in the range 10$^{12}$ -- 10$^{14}$ cm$^{-2}$. The minigap width varies from 1 to 20 meV, depending on the electron concentration. The position of the minigaps in $K$-space (but not their size) is in quantitative agreement with theory.