Fast convolution over small finite fields
A recent paper by Alexey Pospelov appears to give the state of the art.  (It isn't the first to achieve the bounds I'll quote, but it achieves them in a unified way for arbitrary fields, and equally importantly, it states the bounds clearly, see p. 3.)

$\bullet$ We can multiply two degree-$n$ polynomials over an arbitrary field $\mathbb{F}$ using $O(n \log n)$ multiplications in $\mathbb{F}$ and $O(n \log n \log \log n)$ additions in $\mathbb{F}$.  This is originally due to Schonhage-Strassen (for char. $\neq 2$) and Schonhage for char. 2.  As I mentioned, this implies the same bounds for cyclic convolution.  Pospelov also states, 
"We are currently not aware of any algorithms with an upper bound of [the above] that are not based on consecutive DFT applications..."

$\bullet$ Cantor and Kaltofen generalized these results, showing the bounds hold for arbitrary algebras (not just fields).

$\bullet$ If $\mathbb{F}$ supports Discrete Fourier Transform of appropriate order, that is, if $\mathbb{F}$ has a primitive $N$-th root of unity where $N$ is large enough (I believe $N = O(n)$ suffices) and $N$ is a power of 2 or 3, then we can do polynomial multiplication with $O(n)$ multiplications and $O(n \log n)$ additions.  Various other improvements are possible for fields with other special properties.

$\bullet$ It seems to be plausible, but unknown, whether Furer's recent improvement in integer multiplication (reproved in a different way by De et al.) can help lead to faster polynomial multiplication algorithms, over finite fields say.  Can anyone comment?

Todd Mateer's thesis also seems like an excellent resource to understand the FFT literature and applications to polynomial multiplication (thanks Jug!); but you have to dig more to find what you're looking for.