Good question.
Pls. refer to the article by Hansen and Lunde published on JAE: A forecast comparison of volatility models: does anything beat a GARCH(1,1)?. It is classic but old work first done in 2001, although among the literature in 2005. So you'd better check more new progress on this topic.

Any question for the paper, we can talk it later. thanks

I find the paper recommended by 1st floor very good.

Basically, GARCH model is a model that "fits" something, not just for the fat tail or asymmetic process. For homoscedasticity,  and normal distribution in return series, it also performs good. What you need to test is not the fat tail or the skewness, but whether the model fit the data well both in the sample and out of sample. Mainly, out of sample. The paper by the 1st floor gives several criteria to measure the performance.

In short, my point is that, if you do not see any non-zero skewness or high kurtosis by the GARCH model you estimate, it does not mean that GARCH model is not good. It is the data that does not give these characters. If the data has these characters, GARCH model will surely describe these features. What you need to do is just to test the fitness.  

正态性检验；偏度、峰度（与正态分布比较）；平稳性检验；ARCH检验
随便看偏类似的论文，应该都有的吧