The concepts of local and nonlocal models of the breakdown of dense gases are introduced. The basis for a nonlocal model with runaway electrons is discussed. Experimental results on electric discharges in dense gases which develop in a regime of intense electron runaway, in contrast with the classical forms of gas discharges, are reviewed. It is shown that electron runaway plays a fundamental role in the breakdown dynamics of dense gases over a wide range of conditions. Space-time and energy characteristics of the pulses of runaway electrons and of the accompanying $x$ radiation are reported. The involvement of runaway electrons in the breakdown mechanism can be seen in a shift of the minimum on the $U(Pd)$ curves toward higher values of $Pd$ as the overvoltage increases. When the overvoltage reaches a large factor, a polarization selfacceleration, as discussed by Askar'yan, occurs, and runaway electrons with energies $\varepsilon>eU_{\max}$ are generated. The breakdown of a dense gas in a strong field differs from that at a moderate overvoltage in that it ceases to be a purely volume process.