Magmatic liquids, including silicate and carbonate melts, are principal agents of mass and heat transfer in the Earth and terrestrial planets, and they play a crucial role in various geodynamic processes and in Earth's evolution. Electrical conductivity data of these melts elucidate the cause of electrical anomalies in Earth's interior and shed light on the melt structure. With the improvement in high-pressure experimental techniques and theoretical simulations, major progress has been made on this front in the past several decades. This review aims to summarize recent advances in experimental and theoretical studies on the electrical conductivity of silicate and carbonate melts of different compositions and volatile contents under high temperature and pressure. The electrical conductivity of silicate melts depends strongly on temperature, pressure, water content and the ratio of non-bridging oxygens to tetrahedral cations (NBO/T). By contrast, the electrical conductivity of carbonate melts exhibits a weak dependence on temperature and pressure due to their fully depolymerized structure. The electrical conductivity of carbonate melts is higher than that of silicate melts by at least two orders of magnitude. Water can increase electrical conductivity significantly and reduce the activation energy of silicate melts. Conversely, this effect is weak for carbonate melts. In addition, the replacement of alkali-earth elements (Ca2+ or Mg2+) with alkali elements causes a significant decrease in the electrical conductivity of carbonate melts. A distinct compensation trend is revealed for the electrical conductivity of silicate and carbonate melts under anhydrous and hydrous conditions. Several important applications of laboratory-based melt conductivity are introduced in order to understand the origin of high-conductivity anomalies in the Earth's mantle. Perspectives for future studies are also provided.