Time domain integral equation (TDIE)-based methods for the electromagnetic scattering and radiation problems have many potential applications in the areas of high-resolution radar technology, electromagnetic pulse simulation studies, and target identification techniques. These applications could benefit from TDIE methods because of their combination of the strengths of integral equation methods and time domain methods. Specifically, as integral equation methods, they need only surface discretization for homogeneous scatterers, and as time domain methods, they can work for nonlinear problems and can analyze a band of frequencies in a single simulation. Despite these advantages, TDIE methods have historically been inefficient and unstable, and therefore have not been applied broadly. This thesis develops an absolutely stable and accurate TDIE-based technique called the finite difference delay modeling (FDDM) method. In the FDDM method, the temporal discretization is realized by a mapping from the Laplace domain to the [Special characters omitted.] -domain based on a finite difference approximation derived from an ordinary differential equation solution method. Once the system is in the [Special characters omitted.] -domain, it can be inverse-transformed into a discrete time system and solved by marching-on-in-time. For Green's functions with simple Laplace domain expressions, the process can be carried out analytically. For other Green's functions or discretization schemes, a numerical method is employed to calculate the inverse [Special characters omitted.] -transform using trapezoidal rule and discrete Fourier transform (DFT). The first FDDM method developed here computes scattering from perfect electric conductors (PECs). For the temporal discretization, first- and second-order finite difference approximations are used and are shown to be unconditionally stable. For open scatterers, there is a slowly growing, low frequency instability at later time steps because the electric field integral equation is blind to static solenoidal currents which generate no electric field. This problem can be solved by a loop-tree decomposition approach. The second application of the FDDM scheme presented here computes the scattering from homogeneous dielectric bodies. Low frequency instability problems were avoided with another stabilization technique that augments the tangential field boundary condition equations with normal field boundary condition equations. In addition, the FDDM method was applied to dispersive scattering problems. Using FDDM, dispersive scattering is not much harder to model than non-dispersive scattering, though the kernels can be difficult to compute analytically. Thus, a numerical method is employed to compute the inverse [Special characters omitted.] -transform needed to discretize the kernel in time. Finally, to get better temporal convergence, implicit Runge-Kutta based (IRK) based schemes are applied for the temporal discretization. The proposed technique maps a Laplace domain equation to a [Special characters omitted.] -domain equation using the Butcher tableau of the IRK scheme. A discrete time domain system is recovered by computing the inverse [Special characters omitted.] -transform numerically. The resulting technique is capable of third- or fifth-order accuracy in time, and is absolutely stable. Numerical results illustrate the accuracy and stability of the technique.