A fundamental task in the analysis of data is to detect and estimate interesting "structures" hidden in it. In low dimensions, this task has been explored for over 100 years with dozens of developed methods. In this talk I'll focus on aspects of this problem for high dimensional data, where each observed sample has many coordinates, and the number of samples is limited. We will show how in such cases: (i) standard methods to detect structure in high dimensions may not work well ; (ii) how the notion of sparsity can come to the rescue, albeit it brings with it significant mathematical, statistical and computational challenges.