Combinational Logic Circuits and Theory of Computation
Logic circuits are common in complexity theory, where they go by the name circuits. There is a big difference between circuits and models of computation such as the Turing machine: each circuit can only handle inputs of fixed size. In order to fix this, under the circuit computation model, for every input length $n$ there is a circuit $C_n$, and together they compute a function on strings of arbitrary length. This computation model, as stated, is too strong: it can compute uncomputable functions, indeed all functions. The problem is that an infinite sequence of circuits doesn't necessarily have a finite description. In order to fix this problem, we usually demand that the circuits be uniform, that is, that they be generated by some Turing machine, which on input $n$ generates $C_n$.

The circuit model is especially popular in complexity theory, even without the uniformity restriction. The reason is the following observation: a Turing machine running in polynomial time can be converted to a (uniform) sequence of circuits of polynomial size, using essentially the ideas of Cook's theorem (which shows that SAT is NP-complete). Hence if you want to prove that a certain problem cannot be solved in polynomial time, it is enough to show that it cannot be solved by polynomial size circuits.