  This is a summary of the lectures presented at the Xth Brazilian school on
cosmology and gravitation. The style of the text is that of a lightly written
descriptive summary of ideas with almost no formulas, with pointers to the
literature. We hope this style can encourage new people to take a look into
these results. We discuss the variables that Ashtekar introduced 18 years ago
that gave rise to new momentum in this field, the loop representation, spin
networks, measures in the space of connections modulo gauge transformations,
the Hamiltonian constraint, application to cosmology and the connection with
potentially observable effects in gamma-ray bursts and conclude with a
discussion of consistent discretizations of general relativity on the lattice.
