The physics of phase transitions, that happen in the macroscopic world of our everyday life, is pretty well known to us. The most common to our eye is the solid-liquid-gas transition which one can simply observe by tweaking the pressure, volume and temperature. However, when we reduce the temperature to absolute zero, the thermal fluctuation ceases to exist and the carriers experience no resistance. As a consequence the electric current flows undiminished, the vortex in a superfluid spins uniformly forever. In this ultra-cold domain, quantum mechanically driven new exotic phases/phenomena appear in the mesoscopic world (films and wires). The quantum Hall effect is one such phenomenon. In this talk we will discuss how the contemporary physicists Berezinskii, Kosterlitz and Thouless in early 1970s and Haldane in early 1980s used topology, a branch of mathematics, to explain these phases and their transitions. In the last part of the talk I will present recent phenomena like topological insulators, topological superconductors and topological metals.