Elasticity imaging, which is also known as Elastography, aims to determine the elastic property distribution of non-homogeneous deformable solids such as soft tissues. This can be done non-destructively using displacement fields measured with medical imaging modalities, such as ultrasound or magnetic resonance imaging. Elasticity imaging can potentially be used to detect tumors based on the stiffness contrast between different materials. This requires the solution of an inverse problem in elasticity. This field has been growing very fast in the past decade. One of the most useful applications of elasticity imaging may be in breast cancer diagnosis, where the tumor could potentially be detected and visualized by its stiffness contrast from its surrounding tissues. In this work the inverse problem will be solved for the shear modulus which is directly related to the Young's modulus through the Poisson's ratio. The inverse problem is posed as a constrained optimization problem, where the difference between a computed (predicted) and measured displacement field is minimized. The computed displacement field satisfies the equations of equilibrium. The material is modeled as an isotropic and incompressible material. The present work focuses on assessing the solution of the inverse problem for problem domains defined with a continuous and discontinuous shear modulus distribution. In particular, two problem domains will be considered: 1) a stiff inclusion in a homogeneous background representing a stiff tumor surrounded by soft tissues, 2) a layered ring model representing an arterial wall cross-section. The hypothetical "measured" displacement field for these problem domains will be created by solving the finite element forward problem. Additionally, noise will be added to the displacement field to simulate noisy measured displacement data. According to the results of my thesis work, the potential of the elasticity imaging in the medical field is emerging. The inclusion in problem domain 1, representing a stiffer tumor in a uniform background, can be found and located in the shear modulus reconstructions. Thus, these reconstructed images can potentially be used to detect tumors in the medical field. The electronic version of this dissertation is accessible from http://hdl.handle.net/1969.1/155332