The research presented in this thesis was motivated by the desire to understand the flow field within a new digital microfluidic device currently under development. This required an investigation of the dynamics of a droplet migrating along the surface of another fluid due to interfacial surface tension gradients. The quantitative analysis of the flow field presented in this thesis provides the first known solution for the velocity field in a migrating droplet confined to an interface. The first step towards gaining insight into the flow field was accomplished by using the method of reflections to obtain an analytical model for a submerged droplet migrating near a free surface. The submerged droplet model enabled the analysis of the velocity field and droplet migration speed and their dependence on the fluid properties. In general, the migration velocity of a submerged droplet was found to differ dramatically from the classic problem of thermocapillary migration in an unbounded substrate. A boundary-collocation scheme was developed to determine the flow field and migration velocity of a droplet floating trapped at the air-substrate interface. The numerical method was found to produce accurate solutions for the velocity and temperature fields for nearly all parameters. This numerical scheme was used to judge the accuracy of the flow field obtained by the submerged droplet model. In particular, the model was tested using parameter values taken from a digital microfluidic device. It was determined that the submerged droplet model captured most of the flow structure within the microfluidic droplet. However, for a slightly different choice of parameters, agreement between the two methods was lost. In this case, the numerical scheme was used to uncover novel flow structures.