Geometric concepts provide a very fruitful language for quantum (interband) contributions to the dc electrical conductivity of multiband systems. A well-known example is the intrinsic anomalous Hall conductivity, which is based on the Berry curvature. The quantum metric is a second central quantity of band theory but has so far not been related to many response coefficients due to its nonclassical origin. In this talk, I show that the electrical conductivity yields an interband contribution based on the quantum metric. I discuss the implications of this observation in several examples, which range from spiral magnetism in the context of cuprate superconductors to flat-band models. In the former case, the interband contribution due to the quantum metric is crucial for a consistent theoretical description of the Hall number close to the onset of order. In the latter case, interband effects due to the quantum metric can be significantly enhanced and even dominate the conductivity. This is true in particular for topological flat-band materials with nonzero Chern number, where an upper bound exists for the resistivity due to the common geometrical origin of the quantum metric and the Berry curvature. I close by proposing a conductivity minimum due to the quantum metric in low-mobility rhombohedral trilayer graphene.