In this chapter, we will describe the thermocapillary convection, which is induced by a gradient of the surface tension. The first section will be devoted to mathematical background, and in the subsequent sections, development of our understanding of the thermocapillary convection will be described with aid of historical viewpoints. Already since the early twentieth century the buoyancy-driven convection in liquids caused by the thermal expansion was carefully observed and analyzed by the scientists. Later in the mid-twentieth century, the convection due to gradient of the surface tension was noticed in industrial field and scientific analyses were intensively performed. Among them, the emergence of the traveling waves of velocity and temperature fluctuations was observed experimentally and then confirmed theoretically, leading to the designation “hydrothermal wave,” which has attracted both scientific and practical interest. The major tool of the analysis in this field has been,  and remains, the Linear Stability Analysis (LSA), whose fundamental aspects will be explained in this chapter with respect to basic configurations, such as infinitely large planar liquid layers and infinitely long cylindrical columns, as canonical examples. Furthermore, topics like the transition conditions to the oscillatory flows, traveling direction of the waves, and the influence of the heat exchange with environment are also discussed.