Polya–Szego inequalities and, more broadly, symmetrization estimates are fundamental tools in analysis. In this talk, I will discuss their extensions to discrete graphs and survey what is currently known, which, at present, is rather limited. I will also present several directions, questions, and phenomena that appear to be largely unexplored and, in many cases, surprisingly difficult. A substantial part of the discussion will concern (discrete) isoperimetric inequalities and their connection with symmetrization. One of the main aims of the talk is to convey how few general tools are presently available for understanding such inequalities on graphs, despite the fact that their continuum counterparts are classical and form a well-developed theory.