Nonequilibrium conditions are traditionally seen as detrimental to the appearance of quantumcoherent many-body phenomena, and much effort is often devoted to their elimination. Recently this approach has changed: It has been realized that driven-dissipative could be used as a resource. By proper engineering of the reservoirs and their couplings to a system, one may drive the system towards desired quantum-correlated steady states, even in the absence of internal Hamiltonian dynamics. An intriguing category of equilibrium many-particle phases are those which are distinguished by topology rather than by symmetry. A natural question thus arises: which of these topological states can be achieved as the result of purely dissipative Lindblad-type (Markovian) evolution? Beside its fundamental importance, it may offer novel routes to the realization of topologicallynontrivial states in quantum simulators, especially ultracold atomic gases. In this work I go beyond previous studies in giving a general answer to this question in the context of Gaussian ("integer") topological states, concentrating on 2D Chern insulators as the main example. On the one hand I prove a no-go theorem which shows that such dynamics cannot lead to a topological robust (gapped Liouvillian) unique pure steady state as the result of finite-range Liouvillian. On the other hand, I construct a recipe showing that, under the above conditions, a pure topological steady state may result if exponetially-local Liouvillian is allowed. If strictly local evolution is insisted upon, a mixed steady state arbitrarily close to the desired pure state may be obtained. I will also show how such dynamics could be realized with ultracold atoms and similar systems, and how the resulting states may be detected and topologically-classified. Extension to other types of topological insulators and superconductors is also discussed. arXiv:1810.12050v1 [cond-mat.quant-gas]