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Microwave dielectric measurements over the broad bandwith of 10 MHz to 20 GHz were conducted on composites comprising bundles of single-walled carbon nanotubes (SWNT) embedded in an epoxy matrix, in comparison to the nano-graphite and MWNT. It is found that both relative real and imaginary permittivity of the nanocomposites are strong functions of the SWNT concentration, showing large, wide dielectric and electrical response. Distinct resonance around 1.5 GHz is observed at high SWNT concentrations, accompanied by the downshift of the resonance frequency with increasing concentration. Largely, the SWNT-epoxy composites share the behavior of the MWNT owing to structural similarity, much more effective than the nano-graphite. The remarkable, broadband dielectric and electrical properties of the nanotubes acquired in the work originate from their unique seamless graphene architectures, modeled by two major contributions, dielectric relaxation/resonance and electronic conduction, which is substantiated by the agreement between theoretical analysis and experimental results. The carbon nanotube composites are prospective for microwave applications and offer experimental evidence for fundamental studies in low-dimensional systems.