In broadest sense, descriptive graph combinatorics is the  study of "definable" graphs on Polish spaces that incorporates the  descriptive set theoretic point of view into the graph-theoretic point  of view. This is usually done by demanding various graph-theoretic  objects such as edge relations, colorings, automorphisms to have  topological/measure-theoretic properties such as being Borel,  projective, continuous, closed and asking to what extent classical  results of graph theory generalize to measurable setting. Over the  last two decades, numerous interesting results have been proven which  demonstrate that this point of view is more than a mere specialization  that lead to fruitful ideas. In the first half of this talk, after  recalling some basic descriptive set theoretic notions, we shall give  a brief overview of some fundamental results in descriptive graph  combinatorics. In the second half of this talk, we will cover some new  results regarding the Borel distinguishing number of Borel graphs. The  results in the second half are from a joint work with Onur Bilge.