Boundary conditions arise in the process of numerical implementation of given physical boundary conditions on the original physical or truncated domain due to restrictions imposed by computational resources. The latter situation is often encountered when an infinite domain is truncated or remapped onto a finite computational domain. Numerical implementation of the physical or numerical interface boundary conditions should preserve the accuracy and stability of the inner numerical method. The inaccuracies and instabilities created by numerical implementation of the interface and boundary conditions may be localized at the boundaries or interfaces, but more often they may propagate through out the whole computational domain. Examples include: (a) transparent boundary conditions for hyperbolic and dispersive systems, (b) Berenger's perfectly matched layer (PML) boundary conditions applied to Maxwell's equations (c) stability analysis in the presence of boundaries and interfaces, and (d) grid interfaces and material interfaces in semi-conductor device simulation models, and finite-difference time-domain (FDTD) discretization of Maxwell's equations.