Eddy current nondestructive evaluation (NDE) is usually carried out by exciting a time harmonic field using an inductive probe. However, a viable alternative is to use transient eddy current NDE in which a current pulse in a driver coil produces a transient field in a conductor that decays at a rate dependent on the conductivity and the permeability of the material and the coil configuration. By using transient eddy current, it is possible to estimate the properties of the conductive medium and to locate and size potential flaws from the measured probe response. The fundamental study described in this dissertation seeks to establish a theoretical understanding of the transient eddy current NDE. Compared with the Fourier transform method, the derived analytical formulations are more convenient when the transient eddy current response within a narrow time range is evaluated. The theoretical analysis provides a valuable tool to study the effect of layer thickness, location of defect, crack opening as well as the optimization of probe design;Analytical expressions have been developed to evaluate the transient response due to eddy currents in a conductive plate based on two asymptotic series. One series converges rapidly for a short time regime and the other for a long time regime and both of them agree with the results calculated by fast Fourier transform over all the times considered. The idea of asymptotic expansion is further applied to determine the induced electromotive force (EMF) in a pick-up coil due to eddy currents in a cylindrical rod;Starting from frequency domain representation, a quasi-static time domain dyadic Green's function for an electric source in a conductive plate has been derived. The resulting expression has three parts; a free space term, multiple image terms and partial reflection terms. The dyadic Green's function serves as the kernel of an electric field integral equation which defines the interaction of an ideal crack with the transient eddy currents in a conductive plate. The crack response is found using the reciprocity theorem. Good agreement is observed between the predictions of the magnetic field due to the crack and experimental measurements.