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This book presents the perfectly matched layer (PML) absorbing boundary condition (ABC) used to simulate the surrounding free space when solving the Maxwell equations with such finite methods as the finite difference time domain (FDTD) method or the finite element method. The frequency domain and the time domain equations are derived for the different forms of PML media, namely the split PML, the CPML, the NPML, and the uniaxial PML, in the cases of PMLs matched to isotropic, anisotropic, and dispersive media. The implementation of the PML ABC in the FDTD method is described with details. Propagation and reflection of waves in the discretized FDTD space are derived and discussed, with a special emphasize on the problem of evanescent waves. The optimization of the PML ABC is described for two typical applications of the FDTD method, firstly wave-structure interaction problems, secondly waveguide problems. A review of the literature on the application of the PML ABC to other numerical techniques of electromagnetics and to other partial differential equations of physics is provided