"Discrete logarithm" is a wide class. Originally, this means that we work in a finite field (e.g. integers modulo a big prime) and, given g, p and gx mod p, it is computationally difficult to recover x (it becomes impossible with today's technology once p is big enough).
At some point, someone noticed that discrete logarithm was a special case of a larger situation, applied to a generic group. As such, we could use other kinds of group, in particular elliptic curves. Confusingly, this is also called "discrete logarithm" although the involved mathematics are quite distinct. There are generic algorithms for breaking discrete logarithm, which work on all groups but are expensive; and there are faster algorithms which only work in the original discrete logarithm group (integers modulo a big prime).
There is a fairly general theorem due to (I think) Gilles Brassard, which says that any hard NP problem (in short words, a problem for which no fast solving algorithm is known, but for which a given solution can be verified efficiently) is amenable to being turned into a zero-knowledge proof, which, when turned into a non-interactive zero-knowledge proof, can become a digital signature algorithm. The trick, however, is to find a problem such that the corresponding signature algorithm is tolerably fast; Brassard's theorem is about asymptotic behavior: given sufficiently large parameters, there is a wide enough performance difference between using the algorithm and breaking it, but nothing guarantees that the "using" part is actually doable on a computer which exists right now.
@PulpSpy talks about lattices and coding theory. The subset sum problem (also called "knapsack") is traditional but all the fast variants were broken by lattice reduction; one remaining unbroken knapsack cryptosystem is due to Naccache and Stern, but that algorithm uses multiplications where the older schemes used additions, and this makes it quite unattractive, performance wise. Another kind of problem which looks promising is multivariate quadratic equations; the corresponding asymmetric algorithms being HFE (encryption) and Quartz and Sflash (signature). They potentially allow very small signatures (e.g. 128 bits) but their security is still debated.