This dissertation presents matched interface and boundary time-domain (MIBTD) methods for solving both transverse magnetic (TM) and transverse electric (TE) Maxwell's equations in non-dispersive and dispersive media with complex interfaces and discontinuous wave solutions. In this thesis, five following problems will be discussed: (1) Dielectric interface problems; (2) Debye dispersive interface problems in TM mode; (3) Drude dispersive interface problems in TM mode; (4) Debye dispersive interface problems in TE mode; and (5) Perfectly matched layer (PML) boundary conditions for dispersive interface problems. It is well known in the electromagnetic interface problems that field components across the interfaces are often nonsmooth of even discontinuous. Consequently, the finite-difference time-domain (FDTD) algorithms without a proper interface treatment will cause a staircasing error when dealing with arbitrary interfaces; and only first-order of accuracy is achieved by those FDTD methods. Thus, to restore the accuracy reduction of the collocation FDTD approach near an interface, the physical jump conditions relating discontinuous wave solutions on both sides of the interface must be rigorously enforced. For this purpose, a novel matched interface and boundary (MIB) scheme is proposed to handle material interface problems, in which new jump conditions are derived so that the discontinuous and staggered features of electric and magnetic field components can be accommodated. That results in the staircasing error is totally eliminated in the dielectric interface problems. However, in the dispersive materials like Debye media, interface conditions are now time-dependent. Thus, interface auxiliary differential equations (IADEs) are utilized to describe the transient changes in the regularities of electromagnetic fields across a Debye dispersive interface. In addition, in TM mode, to assist the track of the jump condition information along the interface, a novel hybrid system, which couples the wave equation for the electric component with Maxwell's equations for the magnetic components, is constructed based on the auxiliary differential equation (ADE) approach. As a result, the staircasing error is also eradicated for the Debye interface problems. However, this MIBTD approach is only designed for Debye material equations formed by first-order ADE. Because of that, the MIBTD algorithm for the problem (2) cannot be directly extended to solve Drude dispersive interface problems having second-order ADE. To achieve high order accuracy for the problem (3), a novel hybrid Drude-Maxwell system and IADEs are also formulated to update the regularity change of the field components across interfaces so that the staircasing error is free in the numerical results. In the dispersive interface problems in TE mode, the jump conditions of the electric components become more complicated than in the TM mode case, and rigorously depend on the unknown flux density fields. Therefore, the standard Maxwell's equations are taken into consideration instead of the hybrid system. The leapfrog scheme is employed to simplify the complexities of the jump conditions' derivations in the TE mode, whereas the fourth-order Runge-Kutta method is exploited in the other cases. In any material interface problems, effective MIB treatments are proposed to rigorously impose the physical jump conditions which are not only time dependent, but also couple both Cartesian directions and different field components. Based on a staggered Yee lattice, the proposed MIB schemes can achieve up to sixth order-accuracy in dealing with the straight interfaces, while the uniform second-order accuracy is always maintained in solving irregular interfaces with constant curvatures, general curvatures, and nonsmooth corners. Based on the numerical verification, our MIBTD algorithms are conditionally stable and more cost-efficient than the classical FDTD methods. Finally, the Berenger's PML is successfully used as absorbing boundary condition (ABC) for the dispersive interface problems. The numerical results are provided to validate the efficiency of that PML ABC.