  A fundamental challenge is to understand nonequilibrium statistical mechanics
starting from microscopic chaos in the equations of motion of a many-particle
system. In this review we summarize recent theoretical advances along these
lines. Particularly, we are concerned with nonequilibrium situations created by
external electric fields and by temperature or velocity gradients. These
constraints pump energy into a system, hence there must be some thermal
reservoir that prevents the system from heating up. About twenty years ago a
deterministic and time-reversible modeling of such thermal reservoirs was
proposed in form of Gaussian and Nose-Hoover thermostats. This approach yielded
simple relations between fundamental quantities of nonequilibrium statistical
mechanics and of dynamical systems theory. The main theme of our review is to
critically assess the universality of these results. As a vehicle of
demonstration we employ the driven periodic Lorentz gas, which is a toy model
for the classical dynamics of an electron in a metal under application of an
electric field. Applying different types of thermal reservoirs to this system
we compare the resulting nonequilibrium steady states with each other. Along
the same lines we discuss an interacting many-particle system under shear and
heat. Finally, we outline an unexpected relationship between deterministic
thermostats and active Brownian particles modeling biophysical cell motility.
