A fundamental task in (unsupervised) analysis of data is to detect and estimate interesting "structure" 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 observation has many coordinates. We will show that (i) standard methods to detect structure in high dimensions may not work well, and there is a need to devise new models and approaches; and yet, (ii) with a limited amount of high dimensional observations, regardless of the method employed, there are fundamental limitations to the ability to detect various structures, even if they are present in the data.