  In recent years, there is an increasing interest in transport phenomena that
are fundamentally linked to the presence of multiple bands. In this thesis, we
develop, discuss, and apply a theory of the electrical conductivity that
includes interband contributions within a microscopic approach. We derive
formulas of the conductivity tensor $\sigma^{\alpha\beta}$ and the Hall
conductivity tensor $\sigma^{\alpha\beta\eta}_\text{H}$ for a general two-band
model. This minimal model of a multiband system captures a broad variety of
very different physical phenomena ranging from spiral spin density waves to
Chern insulators. We motivate and derive a unique and physically transparent
decomposition of the conductivity tensors by identifying intra- and interband
contributions as well as symmetric and antisymmetric contributions under the
exchange of the current and the electric field directions. Using these
criteria, we find that the symmetric interband contribution of
$\sigma^{\alpha\beta}$ is connected to the quantum metric, whereas the
antisymmetric part involves the Berry curvature and captures the intrinsic
anomalous Hall effect. We include a phenomenological relaxation rate $\Gamma$
of arbitrary size to study the relevance of the interband contributions
systematically.
  Our conductivity formulas are applied to models and experiments of recent
interest. We identify typical scaling behaviors of the conductivities with
respect to $\Gamma$ in agreement with theoretical and experimental results.
Recent experiments on hole-doped cuprates under very high magnetic fields show
a drastic change of the Hall number when entering the pseudogap regime, which
is shown to be consistent with the onset of spiral magnetic order. We analyze
the experimental results with our formulas of the longitudinal and the Hall
conductivity and clarify the validity of the broadly used Boltzmann-like
conductivity formulas.
