We investigate the role of nondeterminism in Winfree's abstract tile assembly model, which was conceived to model artificial molecular self-assembling systems constructed from DNA. By nondeterminism we do not mean a magical ability such as that possessed by a nondeterministic algorithm to search an exponential-size space in polynomial time. Rather, we study realistically implementable systems that retain a different sense of determinism in that they are guaranteed to produce a unique shape but are nondeterministic in that they do not guarantee which tile types will be placed where within the shape. We show a “molecular computability” result: there is an infinite shape S that is uniquely assembled by a tile system but not by any deterministic tile system. We show a “molecular complexity” result: there is a finite shape S that is uniquely assembled by a tile system with c tile types, but every deterministic tile system that uniquely assembles S has more than c tile types. In fact we extend the technique to derive a stronger (classical complexity theoretic) result, showing that the problem of finding the minimum number of tile types that uniquely assemble a given finite shape is ΣP2-complete. In contrast, the problem of finding the minimum number of deterministic tile types that uniquely assemble a shape is NP-complete [5].