Advanced procedures using the Median statistic With the risk of asking a duplicate question (to be fair, I looked at 10 pages):
Could you name and explain some advanced procedures which make use of the median statistic? 
Background: I am interested because today I saw a presentation on a TAR model, trying to figure out a baseline for negative affect within observants. The mean was taken as frequentist approach and also a Bayesian approach was done. My point was that median might be better, since if for some reason a person, when happy, is REALLY happy, while when sad would not be so different from a baseline. The mean would then not be a good estimate and I figured the median would be better?
EDIT: Maybe easier to comprehend: For which computations are medians used beyond simple descriptive statistics (and why is better than taking the mean for example)?
 A: Quantile regression might be an example of what you are looking for.  Traditional least-squares regression is implicitly approximating the conditional mean of your predicted variable, but quantile regression is instead approximating the conditional median (or really any desired quantile).  By not squaring the deviations, quantile regression treats outliers differently than techniques which rely upon squaring the deviations.  Of course whether this is a good thing or not depends upon what one is trying to do.
It's tough to tell, but I think perhaps in your case you are dealing with a scenario involving some asymmetric loss function.  Common loss functions (that often have nice mathematical properties) often treat overestimates the same as underestimates, but this need not be the case in a particular domain.  Asymmetric loss functions can reflect this reality.
A: As indicated by @gung, the way your question is asked, there can be many answers. 
Of course that can be an opportunity and let me take it. 
First you will likely will want to decide if it’s the median of the probability generating process you think should be modelled (REALLY happy comment) as opposed to using the  sample median to learn whatever you can about the  probability generating process (non-parametric ways comment). 
I’ll focus on the second and start with a situation (that my Phd thesis was about) where you have no choice - because rather than getting the actual data you just get a summary of it from someone else. Bayes is immediate as you now condition the prior on that observation you got  (the sample median) which is all you have and the only tricky part is working out the likelihood for some assumed probability data model  - which is trivial if the number in the sample was odd – 
p(x=sample median) * p( trunk(n/2) of x,s < sample median) *  p( trunk(n/2) of x,s > sample median). 
From the frequentist perspective you can use the same likelihood or some approximation it. So it can apply to any statistical model and usually there is more to the summary than just the median. 
Now if you are interested, Steve MacEachern has started looking at choosing to just use robust summaries in Bayesian analyses even when you do have all the raw data. You should be able to find an abstract from this summer’s Joint  Statistical Meetings in San Diego.
Partial answer to the question below from @Dualinity:
Actually, no because the authors had taken one of those introductory stats courses (and maybe even checked here) and became convinced that the mean was an invalid summary to provide. Now there is a rigorous way to rank sample summaries with respect to capturing the information content of a sample and here the mean will beat the median for most assumed probability models. 
But even some statisticians are so convinced the median has better properties that when I started to go into this in a seminar, the senior statistician dismissed the arguments before I gave them and starting talking to people around him about other things.  I have perused some of the related entries and they seem to miss the distinction between population parameters and sample statistics (e.g. why it’s not always the case that the sample median is the best estimate of the population median) as well as distinguishing the purposes (e.g. describing the population or sample versus inference or learning about a population). 
Fisher said it best a long time ago: “Best for what?”
