Hadronic light-by-light scattering contribution to the muon g-2 on the lattice

We briefly review several activities at Mainz related to hadronic light-by-light scattering (HLbL) using lattice QCD. First we present a position-space approach to the HLbL contribution in the muon g-2, where we focus on exploratory studies of the pion-pole contribution in a simple model and the lepton loop in QED in the continuum and in infinite volume. The second part describes a lattice calculation of the double-virtual pion transition form factor $F_{pi^0 \gamma^* \gamma^*}(q_1^2, q_2^2)$ in the spacelike region with photon virtualities up to 1.5 GeV$^2$ which paves the way for a lattice calculation of the pion-pole contribution to HLbL. The third topic involves HLbL forward scattering amplitudes calculated in lattice QCD which can be described, using dispersion relations (HLbL sum rules), by $\gamma^* \gamma^* \to$ hadrons fusion cross sections and then compared with phenomenological models