 Neural networks have sometimes been discussed as computing systems that may go beyond Turing-computability (e.g., Smolensky 1988). If we restrict our attention to classes of neural networks that contain all systems with current or foreseeable practical applications, this opinion is unwarranted. There is now a considerable literature on the computational and complexity properties of large classes of neural networks. The most relevant systems have digital inputs and outputs (so as to satisfy our first constraint on physical computation) but may have, and typically do have, non-digital internal processes (i.e., their internal processes are not discrete step-by-step manipulations of strings of digits). The main results are the following: (i) feedforward networks with finitely many processing units are computationally equivalent to Boolean circuits with finitely many gates; (ii) recurrent networks with finitely many units are equivalent to finite state automata; (iii) networks with unbounded tapes or an unbounded number of units are equivalent to TMs (Šíma &amp; Orponen 2003).