This introductory Chap. 1 is a survey of one aspect of the mathematical theory of combustion. This book is devoted to the study of elliptic boundary value problems arising in combustion theory which obey a general Arrhenius equation and a general Newton law of heat exchange. We prove that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless rate of heat production (the Frank-Kamenetskii parameter). Moreover, we give sufficient conditions for our semilinear boundary value problems to have three positive solutions, which suggests that the bifurcation curves are S-shaped. In a reacting material undergoing an exothermic reaction in which reactant consumption is neglected, heat is being produced in accordance with Arrhenius rate law and Newtonian cooling. Thermal explosions occur when the reactions produce heat too rapidly for a stable balance between heat production and heat loss to be preserved. In this book we are concerned with the localization of the values of a dimensionless heat evolution rate at which such critical phenomena as ignition and extinction occur. The contents of this monograph are divided into five parts.