The practice of using Gaussian copulas in modeling credit derivatives has come under a lot of criticism in the past few years. What are the major arguments against using the copula method in this respect?
The limitations of the Gaussian copula were well-known among the quantitative finance practitioners before the crisis. See this paper by D. Brigo.
To answer the question:
This said, all other models are either worse or offer cosmetic improvements. Changing the Gaussian factors to some others doesn't really give you much. A few years ago the Random Factor Loading model was en vogue, but it turned out to be much harder to calibrate, and still not flexible enough.
If you want a 'pop science' account for it, the Wired article by Felix Salmon is a pretty good start.
If you want harder technical stuff, well then you can start at the Wikipedia article and its section on Applications and follow the references: