Dahlen's work on the behaviors of finite-frequency seismic waves opened a new field, in which the propagation of waves and its relation to the physical properties of the media are more accurately represented than in previous tomographic studies. The important issues we now face are how to harness the power of the finite-frequency theory, fully extract rich information about the earth from three-component broadband records, and obtain accurate images of the earth's interior. I will present several recent developments in finite-frequency sensitivity kernels for 3D, heterogeneous reference models based on the scattering-integral methods. I will show that the different components of the same arrival at the same receiver have different traveltime and amplitude sensitivities to variations in the velocity structure. This is attributed to scattered waves that affect the waveforms on the different components by various amounts. Furthermore with synthetic, 3D cross-well tomography, we demonstrate that the choice of the coordinate system in which travel times and amplitude anomalies are measured affects the Frechet sensitivity kernels and results in significant differences in tomographic resolution. To date the inversion of P and S wave-speeds have been carried out separately under the assumption that P traveltimes are affected only by the P wave-speed of the elastic media and S traveltimes by the S wave-speed. However, for finite-frequency waves, S wave-speed perturbations may have significant effects on P waveforms. So when waveform-derived traveltime and amplitude anomalies are used in tomographic inversions, the P-wave measurements should be related to not only P wave-speed perturbations but also S wave-speed perturbations.