I am a bit confused as to the conditions under which I can use ttests and ANOVA. I keep reading that the assumption is that the data are normally distributed but does this apply to the population data or the sampling distribution for the sample means?

My understanding of the Central Limit Thereom is that once the sample size gets to about 30 then the sampling distribution will approximate to normal. The data I have is distinctly non-normal (skewed to the right) and all my sample sizes are above 30. Does this mean I can or can't use ttest s and ANOVA since my original data is non-normal but the sampling distribution will be approximately normal (as n>30)?