Focus on some nonequilibrium issues

•We review some relevant aspects of modern nonequilibrium statistical mechanics.•We shed light on the recently introduced t-mixing condition.•We discuss linear response theory for overdamped diffusion processes.

A mathematical framework for the physics of nonequilibrium phenomena is gradually being developed. This review is meant to shed light on some aspects of response theory, on the theory of Fluctuation Relations, on the so-called t-mixing condition, and on the use of large deviation techniques in the description of stochastic diffusion processes.