Electrodeposition is a widely practiced method for creating metal, colloidal, and polymer coatings on conductive substrates. In the Newtonian liquid electrolytes typically used, the process is fundamentally unstable. The underlying instabilities have been linked to failure of microcircuits, dendrite formation on battery electrodes, and overlimiting conductance in ion-selective membranes. We report that viscoelastic electrolytes composed of semidilute solutions of very high–molecular weight neutral polymers suppress these instabilities by multiple mechanisms. The voltage window ΔV in which a liquid electrolyte can operate free of electroconvective instabilities is shown to be markedly extended in viscoelastic electrolytes and is a power-law function, ΔV : η1/4, of electrolyte viscosity, η. This power-law relation is replicated in the resistance to ion transport at liquid/solid interfaces. We discuss consequences of our observations and show that viscoelastic electrolytes enable stable electrodeposition of many metals, with the most profound effects observed for reactive metals, such as sodium and lithium. This finding is of contemporary interest for high-energy electrochemical energy storage.