  The paper is devoted to the problem of finding explicit combinatorial
formulae for the Pontryagin classes. We discuss two formulae, the classical
Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces
and the local combinatorial formula obtained by the author in 2004. The latter
formula is based on the notion of a universal local formula introduced by the
author and on the usage of bistellar moves. We give a brief sketch for the
first formula and a rather detailed exposition for the second one. For the
second formula, we also succeed to simplify it by providing a new simpler
algorithm for decomposing a cycle in the graph of bistellar moves of
two-dimensional combinatorial spheres into a linear combination of elementary
cycles.
