Recovering data from compressed number of measurements is ubiquitous in applications today. Among the best know examples are compressed sensing and low rank matrix recovery. To some extend phase retrieval is a prominent such example. The general setup is that we would like to recover a data point lying on some manifold having a much lower dimension than the ambient dimension, and we are given a set of linear measurements. The number of measurements is typically much smaller than the ambient dimension. So the questions become: Under what conditions can we recover the data point from these linear measurements? If so, how? The problem has links to classic algebraic geometry and other areas of mathematics. In this talk I'll give a brief overview and discuss some of the recent progresses.