The topology of quantum materials is the frontier research in condensed matter physics. In contrast with the conventional classification of materials by using the local symmetry breaking criterion, the states of quantum systems are classified according to the topology of wave functions. The potential applications of topological states may lead the traditional microelectronics to break through and accelerate the significant improvement in topological electronics. Most of the recent studies focus on the topological states of quantum systems under equilibrium conditions without external perturbations. The topological states of quantum systems far from the equilibrium under time-periodic driving have attracted wide attention. Here we first introduce the framework of Floquet engineering under the frame of the Floquet theorem. The nonequilibrium topological states of massless and massive Dirac fermions are discussed including the mechanism of phase transition. Light field driven electronic transition term in the quantum material gains extra time-dependent phase. Thereby the manipulation of effective transition term of the electron is realized to regulate the non-equilibrium topological states. We also mention how the photoinduced coherent phonon affects the nonequilibrium topological states of quantum systems from the perspective of atom manufacturing. Furthermore, research outlook on the nonequilibrium topological states is given. This review provides some clues to the design of physical properties and transport behaviors of quantum materials out of equilibrium.