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We combine topology and active matter paradigms in an effort to achieve topology-dictated nonequilibrium self-assembly of topologically distinct active particles. Active colloids are a distinct category of nonequilibrium matter in which energy uptake, dissipation and movement take place at the level of discrete microscopic constituents. They are known to provide types of self-assembly not accessible in traditional condensed matter systems, such as “living crystals” recently studied by Chaikin, Pine and colleagues. However, only topologically trivial types of active colloids have been realized in the past (in other words, the used constituent active particles were spherical or topologically isomorphic to spheres). Our preliminary observations show that the interplay of topologies of surfaces and flow fields generated by the self-propulsion of active particles can result in highly unusual yet controlled and practically useful forms of self-assembly. To pursue this research direction, our group recently succeeded with practical realization of such active topological colloids. The research on this system transcends the traditional disciplines, ranging from mathematics (topological theories), to physics (active particle behavior, self-assembly), and to chemistry and chemical engineering (material synthesis, photopolymerization, and surface functionalization). The future project outcomes may potentially impinge on our ability of designing new nano- and meso-scale materials with properties not encountered in naturally occurring systems.