Thermocapillary-driven convection in a hanging droplet is experimentally investigated under normal-and micro-gravity conditions. A droplet is hung on a heated cylindrical rod facing downward, and another rod cooled is placed just beneath the droplet to create the designated temperature difference between both ends of the droplet. A transition of the flow field from a two-dimensional axisymmetric 'steady' flow to three-dimensional time-dependent 'oscillatory' ones by increasing temperature difference is realized. The oscillatory flow is accompanied with thermal waves due to so-called hydrothermal wave (HTW) instability, which propagates over the free surface at a constant frequency. The present study aims to understand the transition conditions of the flow from the steady to the oscillatory ones, and to characterize the convective field inside the droplet with the HTW by imposing a range of temperature differences. We discuss the transition condition and the flow fields in the oscillatory regimes after the transition obtained in the terrestrial and on-orbit experiments in 'Kibo,' the Japanese Experiment Module aboard the International Space Station. (C) 2018 Elsevier Ltd. All rights reserved.