Understanding of the motion of droplets on surfaces is important in various fields, such as water repellent surfaces, microfluidics, and removal of condensate in thermal managements to name a few. These motions can be induced by a chemical or thermal gradient or simply by gravity. It is well-known that interactions of the drops with surface imperfections give rise to wetting hysteresis that hinder the drop motion. However, this hysteresis can be overcome by agitating the drop with an external noise. In the absence of any hysteresis, the velocity of the drop is simply the product of the acceleration of the applied bias and the Langevin relaxation time, being independent of the strength of noise. With an applied bias on a pinned drop, the main function of the noise is to reduce the effect of hysteresis. The Langevin velocity is reached only at high power of noise when the effect of hysteresis is nearly eliminated. With a bias and a white noise, the drop begins to exhibit a Brownian like motion; however its detailed features and the related work fluctuations differ considerably from the classical Brownian case. These deviations can be attributed to hysteresis, which give rise several non-classical effects including the possible violation of the Stokes-Einstein equation. The drop can also move on a surface in the absence of any net bias, if it is subjected to an asymmetric noise. In this case, hysteresis is absolutely critical for a net motion. However, the response of the drop to an asymmetric noise is non-linear. Thus a non-linear resistive force of the drop (hysteresis plus kinetic friction) and non-linear response of the drop converts internal vibration of the drop to a directed motion. A higher level symmetry breaking leads to a polarized ratchet, in which drops of different sizes move in opposite directions.