The shape evolution of small droplets attached to a conducting surface and subjected to relatively strong electric fields is studied both experimentally and numerically. The problem is motivated by the phenomena characteristic of the electrospinning of nanofibres. Three different scenarios of droplet shape evolution are distinguished, based on numerical solution of the Stokes equations for perfectly conducting droplets. (i) In sufficiently weak (subcritical) electric fields the droplets are stretched by the electric Maxwell stresses and acquire steady-state shapes where equilibrium is achieved by means of the surface tension. (ii) In stronger (supercritical) electric fields the Maxwell stresses overcome the surface tension, and jetting is initiated from the droplet tip if the static (initial) contact angle of the droplet with the conducting electrode is $\alpha_{s}\,{<}\,0.8\pi $; in this case the jet base acquires a quasi-steady, nearly conical shape with vertical semi-angle $\beta \,{\leq}\, 30^{\circ}$, which is significantly smaller than that of the Taylor cone ($\beta_{T}\,{=}\,49.3^{\circ}$). (iii) In supercritical electric fields acting on droplets with contact angle in the range $0.8\pi \,{<}\,\alpha_{s}\,{<}\,\pi $ there is no jetting and almost the whole droplet jumps off, similar to the gravity or drop-on-demand dripping. The droplet–jet transitional region and the jet region proper are studied in detail for the second case, using the quasi-one-dimensional equations with inertial effects and such additional features as the dielectric properties of the liquid (leaky dielectrics) taken into account. The flow in the transitional and jet region is matched to that in the droplet. By this means, the current–voltage characteristic $I\,{=}\,I(U)$ and the volumetric flow rate $Q$ in electrospun viscous jets are predicted, given the potential difference applied. The predicted dependence $I\,{=}\,I(U)$ is nonlinear due to the convective mechanism of charge redistribution superimposed on the conductive (ohmic) one. For $U\,{=}\,O(10kV)$ and fluid conductivity $\sigma \,{=}\,10^{-4}$ S m$^{-1}$, realistic current values $I\,{=}\,O(10^{2}nA)$ were predicted.