  These notes are a didactic overview of the non perturbative and background
independent approach to a quantum theory of gravity known as loop quantum
gravity. The definition of real connection variables for general relativity,
used as a starting point in the program, is described in a simple manner. The
main ideas leading to the definition of the quantum theory are naturally
introduced and the basic mathematics involved is described. The main
predictions of the theory such as the discovery of Planck scale discreteness of
geometry and the computation of black hole entropy are reviewed. The
quantization and solution of the constraints is explained by drawing analogies
with simpler systems. Difficulties associated with the quantization of the
scalar constraint are discussed.In a second part of the notes, the basic ideas
behind the spin foam approach are presented in detail for the simple solvable
case of 2+1 gravity. Some results and ideas for four dimensional spin foams are
reviewed.

General Relativity and Quantum Cosmology;High Energy Physics - Theory