Electromagnetic inverse scattering problems have so many applications in engineering and science that we may say they have the same importance as the direct electromagnetic problems. The goal is to identify different characteristics of an object such as physical shape and permittivity exploiting the scattered fields. The main two categories of applications of inverse electromagnetic problems are nondestructive testing and remote detection. Medical imaging, ground penetrating radars, security detections which are used in geophysics, medical centers, and mines are examples of these applications. Inverse scattering problems can be solved in time or frequency domain. Since calculation and measurement of the scattered fields in frequency domain are easier, solution of the inverse scattering problems in the frequency domain is more common. In general these problems are not well-posed and they are usually nonlinear. This makes them much more complex with respect to the direct problems which are usually linear and well-posed. This complexity is much more severe when the wavelength of the incident wave is comparable with the size of the scatterer. The inverse scattering methods are qualitative or quantitative. The quantitative methods are generally slower but more accurate compared to the qualitative methods. In this thesis, we are concerned about the microwave imaging approach. Conceptually, it is an electromagnetic inverse scattering problem where the goal is to characterize unknown objects from measurements of the scattered fields that result from their interaction with a known incident wave in the microwave frequency range. The majority of the electromagnetic (EM) spectrum can be used for imaging with different operating frequencies providing different information about the object being imaged. Promising results concerning the use of microwave imaging for early detection of breast cancer have been reported recently in the literature where the potential for resolving small tumors has been demonstrated. In security applicatio scanning and surveillance of people cannot use ionizing radiation for safety reasons. The current metal detection solution is limited as it cannot see underneath clothing. Less-intrusive microwave imaging has obvious advantages in being more reliable, covert, and less disruptive to passengers checkpoints. Shape reconstruction of two dimensional perfect electric conductors is a basic inverse electromagnetic problem which its solution methods can be expanded to more complex problems. In this thesis, Keywords: Inverse Scattering, Ill-Posed Problems, Level Set Method, conjugate gradient Method