  We review many-body effects, their microscopic origin, as well as their
impact onto thermoelectricity in correlated narrow-gap semiconductors. Members
of this class---such as FeSi and FeSb$_2$---display an unusual temperature
dependence in various observables: insulating with large thermopowers at low
temperatures, they turn bad metals at temperatures much smaller than the size
of their gaps. This insulator-to-metal crossover is accompanied by spectral
weight-transfers over large energies in the optical conductivity and by a
gradual transition from activated to Curie-Weiss-like behaviour in the magnetic
susceptibility. We show a retrospective of the understanding of these
phenomena, discuss the relation to heavy-fermion Kondo insulators---such as
Ce$_3$Bi$_4$Pt$_3$ for which we present new results---and propose a general
classification of paramagnetic insulators. From the latter FeSi emerges as an
orbital-selective Kondo insulator. Focussing on intermetallics such as
silicides, antimonides, skutterudites, and Heusler compounds we showcase
successes and challenges for the realistic simulation of transport properties
in the presence of electronic correlations. Further, we advert to new avenues
in which electronic correlations may contribute to the improvement of
thermoelectric performance.
