The interfacial dynamics of a viscous droplet immersed in a viscous medium are considered. The characteristic equation is derived using a normal-mode analysis, and solved for arbitrary, finite fluid properties. The cylinder functions in the characteristic equation are solved using a continued fraction algorithm, and the complex decay factor is searched using a modified quasilinearization minimization scheme. Oscillation frequency and damping rate results are presented for various cases of practical interest (liquid-gas and liquid-liquid systems), and the effect of external medium properties are discussed. It is shown that viscosity of the host medium plays an important part in determining the dynamics of the droplet. Numerical results are also compared to exact solutions for limiting cases, and to existing experimental data for both the fundamental and higher-order modes. It is shown that frequency predictions match very well with experiment, and that damping rate predictions underestimate experimental observation in some cases, possibly due to presence of surface impurities. The application of these results to the measurement of surface tension and viscosity of liquid droplets from single-droplet levitation experiments is also discussed. A new inverse method is developed to determine surface tension and viscosity from a knowledge of the frequency of oscillations, damping rate, droplet radius, droplet mass (or density), and mode of oscillations. Results are presented for various modes of oscillations in nondimensional form. Finally, the effect of static deformation and external forces on the oscillations of a liquid droplet are considered with special reference to levitation. For an arbitrary static shape deformation, the frequency spectrum is shown to split into (2l $-$ 1) peaks for a mode l oscillation, and this frequency split is calculated to first order for mode 2, 3, and 4 oscillations. The deformation is then assumed to be a consequence of a general external force, and the frequency split and the static deformation are calculated in terms of the external force parameters. Droplets levitated by acoustic, electromagnetic, and combined acoustic-electromagnetic forces are considered in particular, and it is shown that the effects of asphericity adequately explain the splitting of frequency spectra commonly observed in experiments. The interpretation of spectra with regard to accurate surface tension measurement using the oscillations of levitated droplets is discussed, and the results are applied to some previous experimental results. It is shown that the accuracy of surface tension measurements can improve remarkably if the asphericity caused by the levitating force, and the resultant frequency-split, are taken into account.