The present uncertainty in the ability of Cu to substitute for Al as the conductor material in very large scale integration arises from the perplexing electromigration behavior of Cu interconnects: the electromigration activation energy in multigrained lines is often about two times lower than for grain‐boundary diffusion, while the pre‐exponential factor in the electromigration rate expression is several orders of magnitude smaller than that characteristic of electromigration along grain boundaries. Using literature data, in particular those of drift velocity experiments, we show that regardless of these unusual facts, grain boundaries are still most likely the major electromigration diffusion pathways in Cu interconnects. Based upon recent progress in the theory of grain‐boundary grooving with an arbitrary grain‐boundary flux [Klinger et al. J. Appl. Phys. 78, 3833 (1995)], and the specific model applying the general theory to electromigration [Glickman, Phys. Low‐Dim. Struct. 11/12, 69 (1994)], we explain the major features of electromigration in Cu in terms of the extension of slit‐like grooves along the interconnect line, followed by their merging. Fitting the electromigration activation energy reported for pure Cu into the model suggests that surface diffusion along freshly created groove walls is slow, with an activation energy above 2 eV. Most likely, this is due to trace surface contaminations. Having this new key element in the grain‐boundary grooving model, with surface diffusion acting in effect as a ‘‘healing’’ mechanism rather than as an independent pathway parallel to grain‐boundary diffusion, and using empirical surface diffusion parameters for Cu, enable us to rationalize the major features of Cu electromigration behavior reported in the literature: the values of the electromigration pre‐exponents, the island‐like morphology of the electromigration displacement region, the current density exponent, and the origin of the Sn effect on electromigration parameters. Our main conclusion is that the full advantages of Cu as the most promising electromigration resistant interconnect material can be realized, provided that trace contaminations responsible for suppressing diffusion along freshly created surfaces are identified, controlled and eliminated. This will allow unalloyed, multigrained Cu interconnects to have an electromigration resistance four orders of magnitude larger than Al(2% Cu) at 100 °C. © 1996 American Institute of Physics.