  In this work we describe the Chen-Ruan cohomology of the moduli stacks of
smooth and stable genus 2 pointed curves, and its algebraic counterpart: the
stringy Chow ring. In the first half of the paper we compute the additive
structure of the Chen-Ruan cohomology ring for the moduli stack of stable
pointed genus 2 curves, describing it as a rationally graded vector space. In
the second part we give generators for the even Chen--Ruan cohomology ring as
an algebra on the ordinary cohomology.
