The paper presents an overview of conductive polymer composites based on thermosetting materials, thermoplastics, and elastomers modified with carbon nanotubes (CNTs). To impart conductive properties to polymers, metal, carbon-dispersed materials, or their combinations are used. The inclusion of dispersed materials in polymers is associated with their microstructural features, as well as with polymerization methods. Such polymerization methods as melt mixing, solution technology, and introduction of fillers into the liquid phase of the composite with subsequent polymerization due to the use of a catalyst are known. Polymer composites that are capable of conducting electric current and changing their properties under the influence of an electric field, i.e., having one or more functional purposes, are called “smart” or intelligent. One such application is electric heating elements with the function of adaptive energy consumption or the effect of self-regulation of temperature depending on the surrounding conditions. A wide variety of polymers and dispersed materials with conductive properties determines a wide range of functional capabilities of the composite, including a positive temperature coefficient of resistance (PTCR) required to control temperature properties. The most effective filler in a polymer for obtaining a composite with desired properties is carbon nanomaterials, in particular, CNT. This is due to the fact that CNTs are a nanosized material with a high bulk density at a low weight, which allows for high electrical conductivity. Calculation of model parameters of polymer composites containing carbon nanostructures can be carried out using neural networks and machine learning, which give a fundamentally new result. The article contains sections with an assessment of various types of polymer matrices based on thermosets, thermoplastics, and elastomers. To impart electrically conductive properties, various options for fillers based on Ag, Au, Cu, Ni, Fe, and CNTs are considered. Methods for introducing dispersed fillers into polymer matrices are presented. Functional composites with a positive temperature coefficient and methods for their regulation are considered. The mechanisms of various electrophysical processes in conductive composites are considered, taking into account the resulting electrical conductivity based on the tunnel effect and hopping conductivity. An analysis of electric heaters based on various polymer matrices and dispersed fillers is carried out, taking into account their operating modes. Thus, the conducted review of modern scientific and practical research in the field of obtaining electrically conductive composites based on various types of polymer matrices with nanosized additives allows us to assess the prospects for the formation of functional composites for electrical heating, taking into account the mechanisms of electrical conductivity and new technologies based on machine learning and neural networks.