Driven by the liberation of the latent heat of fusion, a transformation from an amorphous state to the crystalline state may take place in a progressing wave. The process is self-sustaining and is often called "explosive crystallization". In most applications, the crystallization process takes place in thin layers that are mounted on a substrate. In this work, a model for explosive crystallization in a thin amorphous layer on a heat conducting substrate is presented. One to four rate equations are used to describe the kinetics of the amorphous-crystalline transition. For the thin layer, the energy equation is used in a one-dimensional approximation with a heat-loss term. Heat conduction in the substrate is described by introducing a continuous distribution of moving heat sources at the interface. This gives an integral representation for the temperature in the substrate in terms of the unknown source distribution. Provided the substrate's thermal diffusivity is much smaller than the thermal diffusivity of the layer, the integral representation can be inverted and included in the energy equation of the layer. The integral term implies that there is a non-local influence of the temperature distribution in the layer on the heat loss. Optionally, a thermal contact resistance at the interface between layer and substrate is taken into account. The whole process is examined as a wave of invariant shape in a moving frame of reference. A coupled system of one integro-differential equation and one to four ordinary differential equations is obtained. It is solved numerically using a collocation method. The propagation velocity of the wave is obtained as an eigenvalue of the system of equations. Some representative solutions of the system of equations are shown, demonstrating the key features of the process: Typically, the crystallization zone is short compared to the thermal preheating zone in front of the wave. Long crystallization zones and even incomplete crystallization are possible in cases without heat loss. When heat loss is taken into account, the cooling zone behind the wave is long compared to the pre-heating zone. Varying a non-dimensional heat loss parameter, a critical value is found beyond which no crystallization wave of invariant shape is possible. This can also be interpreted as a certain minimum layer thickness. Finally, crystallization-wave velocities are compared with experimental values for explosive crystallization in germanium. Data for 19 different experimental and material parameters are collated from a number of sources. It is necessary to adjust crystallization parameters to achieve a correct magnitude of the wave propagation velocity. For substrate temperatures up to about 700K, the agreement between the analysis and experimental values is reasonable. For larger substrate temperatures, the wave velocity in the experiment remains approximately constant, which is not reflected in the model results. Furthermore, in the experiment, the wave velocity is nearly independent of the layer thickness at a substrate temperature of approximately 775K. This is in accord with our model. Possible sources for the discrepancies between the experimental results and the model are identified and potential areas for future work are discussed.