Processes of heating and melting of paraffin plugs in pipes of oil wells and pipelines by the using of powerful high-frequency (HF) electromagnetic radiation are studied. Evolution of temperature fields in the plugs volumes and melting dynamics for various physical conditions and plugs geometries are investigated. Also studied elimination of paraffin plugs in oil well equipment by alternating switching of an electromagnetic radiation source. The dependence of the dynamics of the plugs melting on the cycle duration and on-off time ratio of a HF-generator is ascertained. The time of initial burning through and complete melting of a plug are determined. A new method of elimination of plugs with the using of a movable source of electromagnetic radiation, the so-called "electromagnetic pig", is proposed. "Electromagnetic pig" eliminates a plug by melting in the process of heating while it absorbs electromagnetic energy. For efficient melting of a dielectric plug the source of electromagnetic radiation moves along the pipeline as of separation surface of solid phase and liquid one is displaces. Estimates of efficiency of the proposed method are made. A numeric modeling of elimination of paraffin films on the internal surface of pipeline by a movable source of electromagnetic radiation is carried out. The time of complete cleaning of a pipeline from paraffin deposits depending on their depth and initial temperature of running oil, power and frequency of HF source are determined.