This paper addresses the problem of moving heat sources, particularly in the context of calculating temperatures at sliding or cutting contacts. Despite the importance of these problems, they have not been systematically studied due to the uncertainty in the nature of the contact and the wide range of numerical parameters. The paper aims to provide a comprehensive discussion of the assumptions and numerical consequences of the mathematical theory for plane sliding. The theory is discussed for uniform plane sources of various shapes moving with constant velocity in a semi-infinite medium with no heat loss from the surface. The paper covers the steady temperatures attained when the motion has been infinite, the buildup of these temperatures, and the temperatures within the medium. It also includes the maximum and average steady temperatures over the area of the source. The paper considers both constant and variable strength sources and a case where the strength and velocity vary over time, such as a temperature flash with a relaxation oscillation. All solutions are exact, and numerical calculations are provided for specific problems. The paper also discusses the application of these theories to the surface temperature of sliding solids, including the case of a semi-infinite slider sliding on a semi-infinite medium and a long square slider with emissivity. Numerical calculations are presented for particular cases, and the results are compared with experimental data. Overall, the paper provides a detailed and systematic approach to the problem of moving heat sources, offering valuable insights and practical applications for engineers and researchers.This paper addresses the problem of moving heat sources, particularly in the context of calculating temperatures at sliding or cutting contacts. Despite the importance of these problems, they have not been systematically studied due to the uncertainty in the nature of the contact and the wide range of numerical parameters. The paper aims to provide a comprehensive discussion of the assumptions and numerical consequences of the mathematical theory for plane sliding. The theory is discussed for uniform plane sources of various shapes moving with constant velocity in a semi-infinite medium with no heat loss from the surface. The paper covers the steady temperatures attained when the motion has been infinite, the buildup of these temperatures, and the temperatures within the medium. It also includes the maximum and average steady temperatures over the area of the source. The paper considers both constant and variable strength sources and a case where the strength and velocity vary over time, such as a temperature flash with a relaxation oscillation. All solutions are exact, and numerical calculations are provided for specific problems. The paper also discusses the application of these theories to the surface temperature of sliding solids, including the case of a semi-infinite slider sliding on a semi-infinite medium and a long square slider with emissivity. Numerical calculations are presented for particular cases, and the results are compared with experimental data. Overall, the paper provides a detailed and systematic approach to the problem of moving heat sources, offering valuable insights and practical applications for engineers and researchers.