This article describes the process of detonation initiation within the scope of the spontaneous flame concept, originally proposed by Zel＇dovich. The basis of the concept is that the origin of certain combustion modes is conditioned by the gradients of self-ignition time delay in the system. Considerable recent attention has been focused on applying this concept to real physical systems. This overview summarizes the essential results of numerical, analytical, and experimental works on the spontaneous detonation initiation performed over the past decade. At present, there are two extreme points of view on the problem under consideration. One is the recognition of the problem from the standpoint of mathematical proof of the appropriate solutions of chemical gas dynamics; and the other is the phenomenological recognition of some combustion processes that cannot be adequately explained within the framework of the laminar flame theory or the classical theory of detonation. Therefore, the mathematical theory appeared separately from specific examples and is sufficiently developed further. Phenomenology was reduced to the analysis of some isolated examples, which have been not supported by the general model except in individual attempts of numerical modeling. At the same time, the analysis does not reject the close connection between investigations performed during last 10 years. To establish such a connection it is necessary to overcame the artificial barriers created by the specificity of the formulations of individual approaches, and to consider the problem of spontaneous regimes of combustion by coupling the above mentioned points of view. Such an attempt seems useful to comprehend the achieved level of understanding and to determine the unresolved or unclear tasks. (C) 2000 Elsevier Science Ltd. All rights reserved.