Cryo-electron microscopy (cryo-EM) is an imagingtechnology that is revolutionizing structural biology, enabling reconstruction of molecules at near-atomic resolution. Cryo-EM produces a large number of very noisy two-dimensional tomographic projection images of a molecule, taken at unknown viewing directions. The extreme levels of noise make classical tasks in statistics and signal processing, such as alignment, detection and classification, impossible. I will start the talk by studying the multi-reference alignment problem, which can be interpreted as a simplified model for cryo-EM. In multi-reference alignment, we aim to estimate multiple signals from their circularly-translated, unlabeled, copies. In high noise regimes, the measurements cannot be alignedor clustered. Nonetheless, accurate estimation can be achieved using non-convex optimization by considering the invarariants of the problem. Then, I will introduce the analog invariants of the cryo-EM problem and focus on two applications, called 2D classification and ab initio modeling.