The paper presents a study of stability of rapidly evaporating droplets and liquid shells occurring in the process of explosive boiling, where, in addition to surface evaporation, a vapor bubble grows within a highly superheated liquid droplet immersed in a liquid or gas medium. To get better insight into the problem, two simpler but related problems are studied before the full stability problem is treated. First, the stability of an evaporating, highly superheated liquid droplet is analyzed, in order to estimate the influence of the outer evaporation from the droplet surface. The linear stability of the process at the final stages of explosive boiling, when the droplet forms an expanding liquid shell, is studied next. Finally, the general case of explosive boiling stability is considered. It is shown that the process is unstable, as indeed has been found in existing experiments. (C) 2001 Elsevier Science Ltd. All rights reserved.