By a modified technique of contact electrical resistance (CER), the electrical conductivity of alcohol and aqueous solutions of acids, salts, and bases in submicron gaps between clean contact surfaces have been measured. Alcohol solutions of water content below 10% do not have substantial electrical conductivity, while at contents of about 50% their electrical conductivity becomes similar to that of aqueous solutions. The electrical conductivity of dilute aqueous solutions is anomalously high. It is higher than that of strong solutions of salts and acids, in contradiction with the ionic theory of electrical conductivity. A conclusion on the nonionic nature of the electrical conductivity of aqueous solutions in submicron gaps is formulated. A model of electrical conductivity assuming the formation of a labile spatial hydrogen-bond network in the contact gap is proposed and used to explain the results.