The proposed work consists of two parts. The first part provides an overview of the classical Schwinger—DeWitt technique for calculating the effective action in quantum field theory and quantum gravity. The background field and heat kernel methods, and the calculation of the heat kernel coefficients for minimal second-order operators are sequentially presented. Then we discuss the application of these methods to the calculation of the divergent part of a one-loop effective action, and, finally, the method of universal functional traces, which is also applicable to higher-order minimal and non-minimal operators. In the second part of the work, we present some new results obtained in recent years on off-diagonal heat kernel expansions for higher-order minimal operators. It is shown that these expansions, which generalize the standard DeWitt's ansatz, have the form of a double functional series in some new special functions, which we call "generalized exponential functions." The properties of these functions and expansions constructed from them are discussed in detail, in particular, the presence in them of terms with arbitrarily large negative powers of proper time. Finally, we describe two different covariant methods for calculating the coefficients of off-diagonal expansions—using the "generalized Fourier transform" and perturbation theory.