Consider a random construction (i.e., a map from sequences of bits to countable structures) that almost surely yields a single structure (or models of a particular theory) up to isomorphism. How much algorithmic randomness is needed in the input sequence to obtain that structure (or models of that theory)? We will examine this question for well-known constructions (such as the Erdős–Rényi random graph) and also consider when we can obtain existing notions of randomness (e.g., Martin-Löf or Kurtz) in this way. Preliminary report on joint work with Nate Ackerman.