Recent progress in observations of Rossby-type waves on the Sun revived the interests towards the waves. Though the hydrodynamic Rossby waves are well studied in the Earth context, ubiquitous existence of magnetic field in the atmosphere/interior of the Sun requires to study the waves in magnetohydrodynamics. The talk will cover the recent direct observations of Rossby waves on the Sun, theoretical development of magneto-Rossby wave theory and the possible connection of the waves to observed long and short term variations in solar magnetic activity.