Nonequilibrium steady states arise if a system is driven in a time-independent way. This can be realized through contact with particle reservoirs at different (electro)chemical potential for enzymatic reactions and for transport through quantum dot structures. For molecular motors, an applied external force contributes to such an external driving. Formally, such systems are described by a master equation with time-independent transition rates that are constrained by the local detailed balance relation. Characteristic of such systems are persistent probability currents. This stationary state is unique and can be obtained either through a graph-theoretic method or as an eigenvector of the generator. These systems have a constant rate of entropy production. Moreover, this entropy production fulfills a detailed fluctuation theorem. The thermodynamic uncertainty relation provides a lower bound on entropy production in terms of the mean and dispersion of any current in the system. An important classification distinguishes unicyclic from multicyclic systems. In particular for the latter, the concept of cycles and their affinities are introduced and related to macroscopic or physical affinities driving an engine. In the linear response regime, Onsager coefficients are proven to obey a symmetry.

Rare or extreme fluctuations beyond the Gaussian regime are treated through large deviation theory for the nonequilibrium steady state of discrete systems and of systems with Langevin dynamics. For both classes, we first develop the spectral approach that yields the scaled cumulant-generating function for state observables and currents in terms of the largest eigenvalue of the tilted generator. Second, we introduce the rate function of level 2.5 that can be determined exactly. Contractions then lead to bounds on the rate function for state observables or currents. Specialized to equilibrium, explicit results are obtained. As a general result, the rate function for any current is shown to be bounded by a quadratic function which implies the thermodynamic uncertainty relation.

The asymmetric random walk is introduced as a simple model for a molecular motor. Thermodynamic consistency imposes a condition on the ratio between the forward and the backward rate. Fluctuations in finite time can be derived analytically and are used to illustrate the thermodynamic uncertainty relation. For the long-time limit, concepts from large deviation theory like a rate function and a contraction can be determined explicitly.