  Combinatorics, in particular graph theory, has a rich history of being a
domain of successful applications of tools from other areas of mathematics,
including topological methods. Here, we survey the study of the Hom-complexes,
and the ways these can be used to obtain lower bounds for the chromatic numbers
of graphs, presented in a recent series of papers
\cite{BK03a,BK03b,BK03c,CK1,CK2,K4,K5}.
  The structural theory is developed and put in the historical context,
culminating in the proof of the Lov\'asz Conjecture, which can be stated as
follows:
  For a graph G, such that the complex Hom(C_{2r+1},G) is k-connected for some
integers r>0 and k>-2, we have \chi(G)>k+3.
  Beyond the, more customary in this area, cohomology groups, the
algebro-topological concepts involved are spectral sequences and
Stiefel-Whitney characteristic classes. Complete proofs are included for all
the new results appearing in this survey for the first time.
