Single particle cryo-electron microscopy (EM) recently joined X-ray crystallography and nuclear magnetic resonance (NMR) spectroscopy as a high-resolution structural method for biological macromolecules. In single particle cryo-EM, the 3-D structure needs to be determined from many noisy 2-D projection images of individual, ideally identical frozen-hydrated macromolecules whose orientations and positions are random and unknown.  I will give a brief introduction to the modern computational challenges in single particle cryo-EM, focusing on 3-D ab-initio modelling and classification of structural variability, and how they can be solved using the mathematics of representation theory, Fourier analysis, convex optimization, semidefinite programming, and dimensionality reduction.