Dielectric materials with high thermal conductivity (TC) can enable disruptive performance enhancement in the areas of electronics packaging, thermal management, energy storage, cabling, and heat sinks. There have been widespread global efforts over the past decade on developing novel nanocomposite dielectric materials. As a baseline, legacy polymers and epoxies used in the abovementioned applications have very low thermal conductivities ranging from 0.1-0.5 W·m-1·K-1. Recent advances have led to the commercial availability of polymeric materials with thermal conductivities approaching 10 W·m-1·K-1. Importantly, several fundamental studies report novel nanocomposites with thermal conductivities > 50 W·m-1·K-1. This article summarizes progress in the development of such materials with a focus on developments that show promise for improved practical dielectrics. This review highlights that high TC alone is inadequate to characterize the suitability of any material for the above applications. Other thermal properties, such as thermal diffusivity, glass transition temperature, and the ratio of the in-plane to out-of-plane TC, are important to quantify the thermal performance of novel nanocomposites. In addition, characterization and understanding of mechanical properties (coefficient of thermal expansion (CTE), tensile strength and elastic modulus) and electrical properties (dielectric strength and dielectric permittivity) are critical for holistic multifunctional assessments of these materials. There are other parameters and properties that influence performance, life, and manufacturability, such as viscosity and moisture absorption. This study reviews all the above aspects of nanocomposite dielectric materials reported in the literature. More specifically, we analyze various filler-polymer combinations, and the influence of approaches to incorporate fillers in the polymer on the thermal, mechanical, and electrical properties. While the addition of fillers leads to huge enhancements in TC, the TC is highly anisotropic, with out-of-plane TC lower than in-plane TC by an order of magnitude. It is seen that most present-day materials are still inadequate for future applications due to their low glass transition temperatures; specific promising materials are highlighted. While the addition of fillers reduces the CTE, further reduction is needed to favorably improve the mechanical performance of these materials. While the electrical insulating properties of these composite materials are adequate, there is very little data reported on other electrical properties. In summary, while there is an understandable focus on enhancing the TC, other properties are underreported, and there is insufficient information to support the assessment of most novel materials for practical applications. Overall, this study summarizes the state-of-the-art dielectric nanocomposites and outlines directions for future research to bridge the gap between basic materials science and applications.