Longitudinal studies are found everywhere in medicine and biology. They are the gold standard of health and nutrition research. Take a look at the nurse’s health study, one of the most important sources of information about health in everyday life as a prime example.
A longitudinal study simply means studying the same group of people more than one time. If I asked you questions this week and again next week, that would count as a longitudinal study. (The converse is a cross-sectional survey, in which you only survey people once.)
One of the difficulties with longitudinal studies that occur over a longer period of time is that people withdraw from the study for many reasons other than death. There are statistical techniques to deal with bias due to attrition. Thankfully, public use datasets typically come with different weights to use based on which wave(s) of the study you’re using, so I’ve never needed to become especially familiar with those techniques!
If you’re interested in longitudinal surveys in the social sciences in the US, you could look at the National Longitudinal Studies or the http://psidonline.isr.umich.edu/. In the UK, there is the British Cohort Study, which has followed everyone born in a certain week in 1970 over the course of their lives. There are a couple of others with older and younger cohorts. I imagine that the websites for these studies will have information about the statistical techniques used to deal with attrition.
Longitudinal surveys are really useful for getting closer to understanding causality than a cross-sectional survey will do, but they are a lot of work.