We study the nonequilibrium time evolution of a variety of one-dimensional (1D) and two-dimensional (2D) systems (including SSH model, Kitaev-chain, Haldane model, p + ip superconductor, etc.) following a sudden quench. We prove analytically that topology-changing quenches are always followed by nonanalytical temporal behavior of return rates (logarithm of the Loschmidt echo), referred to as dynamical phase transitions (DPTs) in the literature. Similarly to edge states in topological insulators, DPTs can be classified as being topologically protected or not. In 1D systems the number of topologically protected nonequilibrium time scales are determined by the difference between the initial and final winding numbers, while in 2D systems no such relation exists for the Chern numbers. The singularities of dynamical free energy in the 2D case are qualitatively different from those of the 1D case; the cusps appear only in the first time derivative. [3,4] are two vividly investigated fields of physics, with no strong bonds between them. The leading role played by topology in condensed matter has only been realized recently with the discovery of topological insulators, the descendants of quantum Hall states. Some of their correlation functions are universal and are not influenced by the microscopic details of the system, but are rather determined by the underlying topology. The analysis of nonequilibrium states, on the other hand, have emerged recently in a different field: in cold atomic systems. With the unprecedented control of preparing initial states and governing the time evolution, a number of interesting phenomena has been observed such as the Kibble Zurek scaling [5], the lack of thermalization in integrable systems [6], etc. In this paper, we connect these two, seemingly unrelated fields and show that topology can be used as an organizing principle to classify out-of-equilibrium systems.The most popular setups for nonequilibrium dynamics are quench experiments in which the quantum system initially sits in the ground state of a given Hamiltonian, but its time evolution is governed by another Hamiltonian. The quench protocol can conveniently be characterized by the dynamical partition function with no reference to any particular observables, defined asFor positive real values of z this gives the partition function of a field theory with boundaries |ψ separated by z [7]. For our purposes, we use z = it with t real, which then gives the Loschmidt amplitude, that is, the overlap of the time-evolved state with the initial state G(t) = Z(it). It characterizes time evolution and the stationary state after a long waiting time [8], and also yields the characteristic function of work [9], which is accessible experimentally [10]. As the time evolution operator is highly nonlocal, it allows Z(z) to be susceptible to the topological properties of the underlying system. Similarly to the equilibrium situation, the dynamical free energy is defined as the logarithm per unit volume f (t) = −1/N d ln G(t). In the thermodynamic li...