Does L have a definition in terms of circuits? Many complexity classes defined with Turing machines have definitions in terms of uniform circuits. For example, P can also be defined using uniform polynomial size circuits, and similarly BPP, NP, BQP, etc. can be defined with uniform circuits.


  So is there a circuit-based definition of L?


An obvious idea would be to allow polynomial size circuits with some depth limitation, but this turn out to define the NC hierarchy.

I was thinking about this question a long time ago, but didn't find an answer. If I remember correctly, my motivation was to understand what the quantum analog of L would look like.
Well, $L = SC^1$, where $SC^1$ is the class of languages computed by polynomial size circuits of $O(\log n)$ width.

As for $NL$, it could be characterized as the class languages computed by polynomial size skew circuits (which in some sense is just another way of saying nondeterministic branching programs).