This paper explores the propagation of electromagnetic waves in materials with indefinite permittivity and permeability tensors, where not all principal components have the same sign. Such materials can exhibit unusual effects like negative refraction, near-field focusing, and high impedance surface reflection. The study shows that a bilayer of these materials can transfer field distributions without internal exponentially growing waves. Recent advances in structured media, such as photonic band gap materials and metamaterials, have expanded the range of electromagnetic material properties. These materials allow solutions to Maxwell's equations not available in naturally occurring materials, leading to new physical phenomena and device development. For example, photonic crystals can modify the radiative density of states, and metamaterials can achieve negative refractive indices. In 2000, it was experimentally shown that a metamaterial with periodic scattering elements could have negative effective permittivity and permeability. This material, previously predicted by Veselago, was demonstrated in a metamaterial composed of wires and split ring resonators. Negative refractive materials have generated interest due to their potential for extraordinary wave propagation phenomena, including near-field focusing. The paper discusses the properties of indefinite media, which can have hyperbolic dispersion relations and support unique wave propagation behaviors. The analysis assumes a linear material with no magnetoelectric coupling, described by permittivity and permeability tensors. Metamaterials can be constructed to closely approximate these tensors. The paper examines the refraction behavior at an interface between vacuum and indefinite media, showing that the group velocity direction depends on the material property tensor. It also discusses the use of compensated bilayers to achieve near-field focusing, with a transfer function that can be made unity for all transverse wave vectors. The paper concludes that indefinite media with indefinite permittivity and permeability tensors have unique properties, including hyperbolic dispersion relations and new mechanisms for sub-diffraction focusing. These materials are not complicated to analyze or fabricate, and their properties have potential applications in various technologies.This paper explores the propagation of electromagnetic waves in materials with indefinite permittivity and permeability tensors, where not all principal components have the same sign. Such materials can exhibit unusual effects like negative refraction, near-field focusing, and high impedance surface reflection. The study shows that a bilayer of these materials can transfer field distributions without internal exponentially growing waves. Recent advances in structured media, such as photonic band gap materials and metamaterials, have expanded the range of electromagnetic material properties. These materials allow solutions to Maxwell's equations not available in naturally occurring materials, leading to new physical phenomena and device development. For example, photonic crystals can modify the radiative density of states, and metamaterials can achieve negative refractive indices. In 2000, it was experimentally shown that a metamaterial with periodic scattering elements could have negative effective permittivity and permeability. This material, previously predicted by Veselago, was demonstrated in a metamaterial composed of wires and split ring resonators. Negative refractive materials have generated interest due to their potential for extraordinary wave propagation phenomena, including near-field focusing. The paper discusses the properties of indefinite media, which can have hyperbolic dispersion relations and support unique wave propagation behaviors. The analysis assumes a linear material with no magnetoelectric coupling, described by permittivity and permeability tensors. Metamaterials can be constructed to closely approximate these tensors. The paper examines the refraction behavior at an interface between vacuum and indefinite media, showing that the group velocity direction depends on the material property tensor. It also discusses the use of compensated bilayers to achieve near-field focusing, with a transfer function that can be made unity for all transverse wave vectors. The paper concludes that indefinite media with indefinite permittivity and permeability tensors have unique properties, including hyperbolic dispersion relations and new mechanisms for sub-diffraction focusing. These materials are not complicated to analyze or fabricate, and their properties have potential applications in various technologies.