Magnetic resonance elastography (MRE) can visualize and measure acoustic shear waves in tissue-like materials subjected to harmonic mechanical excitation. This allows the calculation of local values of material parameters such as shear modulus and attenuation. Various inversion algorithms to perform such calculations have been proposed. Under certain assumptions (discussed in detail), the problem reduces to local inversion of the Helmholtz equation. Three algorithms are considered to perform this inversion: Direct Inversion, Local Frequency Estimation, and Matched Filter. To study the noise sensitivity, resolution, and accuracy of these techniques, studies were conducted on synthetic and physical phantoms and on in-vivo breast data. All three algorithms accurately reconstruct shear modulus, demarcate differences between tissues, and identify tumors as areas of higher stiffness, but they vary in noise sensitivity and resolution. The Matched Filter, designed for optimal behavior in noise, provides the best combination of sharpness and smoothness. Challenges remain in pulse sequence design, delivering sufficient signal to certain areas of the body, and improvements in processing algorithms, but MRE shows great potential for non-invasive in vivo determination of mechanical properties.