  We present a simplified exposition of some classical and modern results on
graph drawings in the plane. These results are chosen so that they illustrate
some spectacular recent higher-dimensional results on the border of topology
and combinatorics. We define a mod2-valued self-intersection invariant (i.e.
the van Kampen number) and its generalizations. We present elementary
formulations and arguments accessible to mathematicians not specialized in any
of the areas discussed. So most part of this survey could be studied before
textbooks on algebraic topology, as an introduction to starting ideas of
algebraic topology motivated by algorithmic, combinatorial and geometric
problems.
