We study the long-time behaviour of interacting particle systems on the integer lattice $\mathbb{Z}^d$ with a focus on symmetries. In particular, we are interested in spontaneous symmetry breaking, i.e., situations where there are stationary measures that satisfy less symmetries than the transition rates themselves. Classical examples include the Glauber dynamics for the Ising model, where the global spin-flip symmetry is broken in dimensions $d\geq 2$ and the translation-symmetry is broken in dimension $d\geq3$. In this talk, we discuss the possibility of a spontaneous breaking of the time-symmetry, i.e., whether an interacting particle system with time-homogeneous rates can admit time-periodic behaviour. While this can easily be ruled out in finite volume, the situation is much more delicate in the thermodynamic limit. We provide a no-go theorem for short-range systems under the additional assumption of reversibility in $d=1,2$ and exhibit examples of non-degenerate interacting particle systems which do indeed exhibit spontaneous time-symmetry breaking. The talk is based on joint work with Benedikt Jahnel.