We compute the integral Grothendieck rings of the moduli stacks of smooth and stable curves of genus two respectively. For the Grothendieck ring of stack of smooth genus two curves, we use the presentation of $\mathcal{M}_2$ as a global quotient stack given by Angelo Vistoli. To compute the Grothendieck ring of stack of stable genus two curves, we first stratify $\overline{\mathcal{M}}_2$ into the boundary divisor $\Delta_1$, parametrizing curves with a separating node, and its complement. Then we use their presentations as quotient stacks given by Eric Larson to compute their Grothendieck rings separately. We show that they are both torsion-free and this, together with the Riemann-Roch isomorphism allows to ultimately give a presentation for the integral Grothendieck ring of stack of stable genus two curves. This talk is based on joint work with Dan Edidin.