  Focus of the chapter is on the theoretical approaches aimed to analyze
thermoelectric properties at the nanoscale. We discuss several relevant
theoretical approaches for different set-ups of nano-devices providing
estimations of the thermoelectric parameters in the linear and non-linear
regime, in particular the thermoelectric figure of merit and the
power-efficiency trade-off. Moreover, we analyze the role of not only
electronic, but also of vibrational degrees of freedom. First, nanoscale
thermoelectric phenomena are considered in the quantum coherent regime using
the Landauer-B\"uttiker method and focusing on effects of energy filtering.
Then, we analyze the effects of many-body couplings between nano-structure
degrees of freedom, such as electron-electron and electron-vibration
interactions, which can strongly affect the thermoelectric conversion. In
particular, we discuss the enhancement of the thermoelectric figure of merit in
the Coulomb blockade regime for a quantum dot model starting from the master
equation for charge state probabilities and the tunneling rates through the
electrodes. Finally, within the non-equilibrium Green function formalism, we
quantify the reduction of the thermoelectric performance in simple models of
molecular junctions due to the effects of the electron-vibration coupling and
phonon transport at room temperature.

Mesoscale and Nanoscale Physics;Strongly Correlated Electrons