normality assumption

When I first learned data analysis, I always checked normality for each variable and made sure they were normally distributed before running any analyses, such as t-test, ANOVA, or linear regression. I thought normal distribution of variables was the important assumption to proceed to analyses. That’s why stats textbooks show you how to draw histograms and QQ-plots in the beginning of data analysis in the early chapters and see if variables are normally distributed, isn’t it?