Abstract:   We review some recent results from the mathematical theory of quantum transport of charge and spin in insulating crystals. The emphasis will be on transport coefficients, such as conductivities. Those are computed as appropriate expectations of current operators in a non-equilibrium almost-stationary state (NEASS), which is driven from an equilibrium state by an external electric field. While for charge transport the usual double-commutator Kubo formula is recovered (also beyond linear response), formulas for appropriately defined spin conductivities are still explicit but more involved. The talk will be based on joint works with G. Marcelli, G. Panati, and S. Teufel.