Various aspects of double-negative (DNG) and single-negative (SNG) metamaterials (MTMs), with the emphasis on the former, as well as combinations of these with conventional double-positive (DPS) materials, have been investigated. The study was initiated by clarifications of specific theoretical aspects associated with DNG materials, and was subsequently extended to investigations of the radiation and scattering from two- and three-dimensional (2D and 3D) MTM-based canonical problems in electromagnetic theory. As to the theoretical aspects of DNG materials, the sign, or more generally the branch, of certain derived parameters associated with them, such as wave number, intrinsic impedance, and refractive index is examined. A consistent set of definitions for these parameters, each yielding a specific sign of the parameter in question, is introduced. It is shown that one definition, and thus the sign, may be more convenient than another, but that the correct solution to a given problem can be obtained with all definitions as long as they are used consistently to provide the expected physical behaviour. These findings are confirmed through a canonical example consisting of a DPS/DNG interface upon which an oblique uniform plane wave is incident. This problem is solved under the assumption that the sign of a given parameter can be positive or negative. It is shown that the use of different signs of the derived parameters leads to different functional forms of e.g., the reflection and transmission coefficients, as well as Snell’s law of refraction, and moreover, that for all choices of the signs, the physical behaviour of the solution is as expected. Previous studies on MTM-based 2D and 3D canonical problems have demonstrated the advantages of exploiting specific arrangements of MTMs in rectangular, cylindrical and spherical configurations to design electrically small, resonant structures such as cavities, waveguides, scatterers and radiators. These ideas are extended here to canonical antenna and scattering configurations which consist of electrically small resonant cylindrical and spherical MTM-based structures excited by an arbitrarily located electric line source and an arbitrarily located and oriented electric Hertzian dipole, respectively. Exact analytical solutions, based on eigenfunction series, are derived and then numerically evaluated to study the radiation and scattering from these structures. Specifically, quantities such as the near field, total radiated power, directivity, as well as the total and differential scattering cross sections, are calculated and their variations with respect to various parameters are investigated. It is shown that the electrically small MTM-based structures possess numerous advantages as compared to the corresponding structures based on the DPS materials alone. In particular, it is demonstrated analytically and numerically, that the cylindrical and spherical MTM-based structures can be designed to be resonant even if their electrical size is arbitrarily small. Significant changes of the field radiated by the respective sources, as well as a significant enhancement of the total radiated power and the scattering cross sections are obtained with these cylindrical and spherical MTM-based structures of which the size is of the order of 1/50 and 1/37, respectively, of the free-space wavelength. The enhancement of e.g., the total radiated power, as compared to the power radiated by the respective sources alone in free space, is found to be of the order of 55 dB, for cylindrical, and 93 dB, for spherical structures. These resonant effects, not present in the corresponding DPS-based structures, are also found to lead to a possibility of controlling the directivity patterns of specific MTMbased structures, as different resonant modes can be excited depending on the location of the source. The influence of losses and dispersion, present in any realistic MTM, has also been investigated. It is demonstrated that the resonant effects of the electrically small MTM-based structures, and thus the resulting enhancements of e.g., the total radiated power, are diminished as the losses are included, while being confined, but still present, to very narrow bands of frequency upon inclusion of dispersion. In addition to these investigations of electrically small structures, the properties of several structures for which the size is comparable to the free-space wavelength have likewise been examined. The results obtained for these structures clearly show there is no advantage in using MTMs in such structures as compared to DPS materials, this being in sharp contrast to the case of electrically small structures.