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We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into ℝ3 is NP-hard. This stands in contrast with the lower dimensional cases which can be solved in linear time, and a variety of computational problems in ℝ3 like unknot or 3-sphere recognition which are in NP and co-NP (assuming the generalized Riemann hypothesis). Our reduction encodes a satisfiability instance into the embeddability problem of a 3-manifold with boundary tori, and relies extensively on techniques from low-dimensional topology, most importantly Dehn fillings on link complements.