Gravity is the curvature of space and time. I will start with a qualitative account of this profound insight of Einstein which underlies his classical theory of General Relativity. While General Relativity provides an exquisitely accurate description of Gravitational physics, its equations break down in extreme physical situations such as those encountered at the first instants of the birth of our Universe and at the centre of Black Holes.
In these situations, it is expected that quantum mechanical aspects of gravity become important. A satisfactory quantum mechanical description of gravity does not yet exist. This, yet unknown, description is called Quantum Gravity. I will discuss the motivations for engaging in the search for Quantum Gravity, and the unique conceptual challenges involved which derive from the fact that the dynamical variable is the very geometry of space and time. Along the way we shall encounter tantalising connections, clues and puzzles thrown up by the strange behavior of Black Holes.

I will end with some remarks about an approach to the problem of quantum gravity which goes by the name of Loop Quantum Gravity.