Some metals share an elusive property with silicon (and other semiconductors): they may exhibit strong self-induced current oscillations during anodic dissolution in electrochemical experiments. While this feature, as well as related features concerning self-organization at reactive solid-liquid interfaces, is still not well understood, the so-called “current-burst model” of the authors succeeded in reproducing many effects quantitatively that have been observed at the Si electrode. The current-burst model assumes that current flow through the electrode on a nm scale is inhomogeneous in both time and space; a single current-burst is a stochastic event. Current oscillations in time and space result from interactions in space or time of single current-bursts. The paper outlines the basics of the model and gives results of Monte Carlo simulations concerning stable and damped oscillations for the current and, as a new feature, for the voltage. With the current-burst model a kind of “nano”-electrochemistry is introduced; its strengths, weaknesses, and possible implications for other electrochemical phenomena and for other materials are briefly discussed.