The understanding of out-of-equilibrium physics, especially dynamic instabilities and phase transitions, is one of the major challenges of contemporary science. Focusing on non-equilibrium dynamics of open dissipative spin systems, I will introduce non-Hermitian Hamiltonian formalism, in which anti-Hermiticity reflects dissipation and presence of non-conservative forces. I will discuss a special case of parity-time-symmetric Hamiltonians that exhibit a phase transition from real to complex eigenspectrum, and how it manifests experimentally. I will then illustrate branch point singularities that appear in the spectrum of some non-Hermitian Hamiltonians, and show how these so-called exceptional points provide access to a wealth of topological effects unique to non-Hermitian systems, e.g. asymmetric transport and non-reciprocal time evolution.