  We provide a gentle introduction, aimed at non-experts, to Borel
combinatorics that studies definable graphs on topological spaces. This is an
emerging field on the borderline between combinatorics and descriptive set
theory with deep connections to many other areas.
  After giving some background material, we present in careful detail some
basic tools and results on the existence of Borel satisfying assignments: Borel
versions of greedy algorithms and augmenting procedures, local rules, Borel
transversals, etc. Also, we present the construction of Andrew Marks of acyclic
Borel graphs for which the greedy bound $\Delta+1$ on the Borel chromatic
number is best possible.
  In the remainder of the paper we briefly discuss various topics such as
relations to LOCAL algorithms, measurable versions of Hall's marriage theorem
and of Lov\'asz Local Lemma, applications to equidecomposability, etc.
