The objective of this work is to apply the method of variation of parameters to various direct and inverse nonlinear, multimode heat transfer problems. An overview of the general method of variation of parameters is presented and applied to a simple example problem. The method is then used to obtain solutions to three specific extended surface heat transfer problems: 1. a radiating annular fin, 2. convective and radiative exchange between the surface of a continuously moving strip and its surroundings, and 3. convection from a fin with temperature-dependent thermal conductivity and variable cross-sectional area. The results for each of these examples are compared to those obtained using other analytical and numerical methods. The method of variation of parameters is also applied to the more complex problem of combined conduction-radiation in a one-dimensional, planar, absorbing, emitting, non-gray medium with non-gray opaque boundaries. Unlike previous solutions to this problem, the solution presented here is exact. The model is verified by comparing the temperature profiles calculated from this work to those found using numerical methods for both gray and non-gray cases. The combined conduction-radiation model is then applied to determine the temperature profile in a ceramic thermal barrier coating designed to protect super alloy turbine blades from large and extended heat loads. Inverse methods are implemented in the development of a non-contact method of measuring the properties and temperatures within the thermal barrier coating. Numerical experiments are performed to assess the effectiveness of this measurement technique. The combined conduction-radiation model is also applied to determine the temperature profile along the fiber of an optical fiber thermometer. An optical fiber thermometer consists of an optical fiber whose sensing tip is coated with an opaque material which emits radiative energy along the fiber to a detector. Inverse methods are used to infer the tip temperature from spectral measurements made by the detector. Numerical experiments are conducted to assess the effectiveness of these methods. Experimental processes are presented in which a coating is applied to the end of an optical fiber and connected to an FTIR spectrometer. The system is calibrated and the inverse analysis is used to infer the tip temperature in various heat sources.