Fair enough. Here's a mini list of sequential issues and fixes for GARCH-type processes, which I'm currently involved with -- and so have on my mind:
Problem: time-varying heteroskedasticity an issue in many financial regressions:
Solution: ARCH(q) regression models (Engle, 1982)
Problem: too many lags, q, in ARCH(q) models make them very unwieldy.
Solution: infinite recursive ARCH lags via GARCH(1,1) (Bollerslev, 1986)
Problem: GARCH(1,1) says market drops and rallies will impact volatility equally.
Solution: GJR-GARCH(1,1) (Glosten, Jagannathan and Runkle, 1993)
Problem: GJR-GARCH(1,1) fits still have too little skewness and kurtosis.
Solution: my fav, among others, GM(K)-GJR-GARCH (GM(K)=Gaussian mixture with K components; Alexander and Lazar, 2006, among others).
Problem: Garch-in-mean effects are ambiguous for all existent GARCH-type models.
Solution: Lewis (forthcoming, 2021), extending GM(K)-GJR-GARCH.
Forthcoming refers to a chapter from my Equity Risk Premium (ERP) book, a current project. My interest is that the 'Garch-in-mean' term is essentially an ERP estimate.