**1. Introduction**

Some of the most spectacular advancements in the description of nature have come from the observation that apparently diverse effects are in reality based on the same physical principles. One of these is the realization that whatever interaction between two entities happens through the exchange of some physical quantity, either momentum, energy, spin, or particles. It is, therefore, of paramount importance to be able to describe how this transfer happens at a nanoscopic level, since these principles are usually fundamental to understand more complex systems, such as materials for energy applications, biological systems, and so on [1–4]. For example, it is now clear that our current inability in reproducing solar energy conversion that takes place normally in plants is related to the difficulties in understanding how light is transformed into an electrical current inside a leaf, in particular how the carriers are split and then move away from the light-receptor centers. At the same time, one key point in this energy conversion process is how excess energy is dissipated. Nature has developed for the leaf feedback mechanisms avoiding it being burnt if storing too much energy—these self-regulating mechanisms are ye<sup>t</sup> outside our understanding and only recently with the advancement of both theoretical and experimental methods have some breakthroughs have been made [5,6].

In this review, we will discuss some of the most common approaches of widespread use in the nanoscale transport community. Here, we assume that a "device" of size ranging from 1 to a few hundreds nm is connected to some macroscopic metallic leads that can induce electrical and energy transfer through the device. We will focus on two fundamental models to describe this transport: on the one hand we will discuss the Landauer's formalism for quantum transport. This is valid when the flow crosses the device essentially without being scattered, i.e., in the so-called ballistic regime: an accurate description when the mean free path of the particles, i.e., the distance between two scattering events, is larger than the dimensions of the device. On the other hand, we will consider the Boltzmann's equation for transport. Here, we assume particle are effectively described by a distribution function and they can be scattered by particle–particle interaction, impurities, or by the device. In this case, the mean free path must be small, otherwise the description through an out-of-equilibrium distribution function does not hold. Both theories can be equally applied to the problem of energy and electric current transport, i.e., to the description of the dynamics of electrons and phonons. The ability of treating on essentially equal footing electrons and phonons allows for a comprehensive description of the device maintaining somewhat a consistent level of accuracy. In general, the outcome of the theory are the transport coefficients [1,2,4]. They serve in describing how the device responds to external stimuli such as a thermal gradient or a bias voltage. We will therefore focus on the electrical conductance *σ*, the Seebeck's coefficient *S*, and the electron and phonon thermal conductances *κe* and *<sup>κ</sup>p*, respectively.

After having introduced the fundamental models, we will discuss some of their uses, generalizations, and other methods applicable to the intermediate regime where the mean free path is comparable with the dimension of the device. In particular, we will discuss the role of electron–electron and electron–phonon coupling in modifying the transport coefficients. The review is organized as follows: Section 2 introduces Boltzmann's equation and Landauer's quantum transport formalisms. We will discuss their formulation using the most advanced electronic structure methods and discuss critically their advantages and limits. This section also allows us to introduce a consistent notation for the rest of the presentation; In Section 3 we will discuss some recent attempts to go beyond the standard approaches. We will explore, for example, the role of electron–phonon interaction and strong electron correlation in affecting the transport coefficients, and introduce theories and models that allow description of these effects efficiently while maintaining the strength of the general formalisms; Finally, Section 4 contains an outlook of some potential lines of further investigation.

### **2. Static Approaches: Semiclassical and Quantum Transport Approaches**

The calculation of the transport coefficients of a nanoscale device requires an accurate description of both electron and phonon (or, whenever translational invariance is broken, vibrational) properties. This description is used into standard methodologies to evaluate the transport coefficients and afterwards, e.g., the figure-of-merit of thermoelectric energy conversion. In this review, we will describe two of these methods, namely the Boltzmann's transport theory which can be seen as a semiclassical method since it is based on the evaluation of the distribution function through velocities and density of states of the device, and the Landauer's approach to quantum transport, which on the other hand describes a ballistic particle transfer through scattering between states in the leads [1,2,4]. There are many Approaches, however, of high scientific significance due to their accurate description of particle–particle interaction that we will not discuss here. One of these is the rate equations formalism [7], which can be made extremely accurate and describe strongly correlated system, but its wild scaling with the number of states reduces its applicability to either simplified models or small systems.
