Shear Wave Elastography (SWE) involves estimating mechanical properties through inversion, i.e., matching measured and simulated propagation characteristics of shear waves in the tissue. The accuracy of the estimated properties depends significantly on the specific characteristics/responses that are being matched. These could range from simple group velocity to dispersion curves and to full-wave response (particle velocity measurements). Using specific applications of arterial, liver, and tumor elstography, we illustrate that effective SWE is performed by resorting to an inversion approach, or combination of inversion approaches, guided by the underlying physics. To this end, we present inversion approaches ranging from matching dispersion characteristics to matching full waveform responses and provide rationale for choosing the appropriate technique(s) depending on the problem at hand.