We describe recent advances in our understanding of the analytic structure of Feynman integrals. In particular, we describe two new classes of constraints on such integrals, that identify discontinuities that either cannot be repeated, or that always give rise to the same result (no matter which other discontinuities are computed first). These new constraints hold at all orders in dimensional regularization, and provide us with new input for the Landau bootstrap, where information about the singularities and discontinuities of individual Feynman integrals is used to construct their functional form.