Eddy current nondestructive testing (NDT) methods are extensively used in the inspection of aging aircrafts. Thousands of fasteners and bonded joints on each aircraft should be inspected and in order to handle the huge task, fast, accurate and cost effective inspection methods are clearly needed. Some of the challenges encountered in eddy current testing are (1) detection of corrosions or cracks in the multi-layer structures, (2) detection of cracks under the fastener (CUF), (3) detection of surface and subsurface defects close to edges.;Conventional eddy current inspection method is time consuming due to the small probe size and large inspection area. Furthermore it requires trained operators for data interpretation. Development of new techniques for rapid and accurate inspection is of considerable interest to the aviation industry. In this dissertation, a more recent eddy current technique called magneto-optic imaging (MOI) is studied.;The availability of a theoretical model that can simulate the MOI system performance is extremely important for understanding and optimizing the MOI sensor and hardware system. In this dissertation, finite element (FE) methods have been applied to numerically compute the electromagnetic fields associated with MOI testing with respect to variation in parameters such as operating frequency, source current and sensor parameters. Most of the testing geometries are three-dimensional and consequently the FE models require extensive computational resources. This paper presents a robust FE model based on AN formulation and a fast iterative solver, which offers a distinct advantage in terms of computational time and data storage. A major contribution of this work is the development of Element-Free Galerkin (EFG) method for NDE applications. In order to model the complex CUF geometry, EFG method has been studied. Two and three-dimensional models for Poisson equation and diffusion equation, describing static or low frequency problems, have been developed and compared with conventional FE methods. EFG methods have also been applied to multi-frequency and pulsed eddy current problems. Simulation results clearly demonstrate the feasibility of the method for time-dependent field applications.