We will formulate several data-driven applications as MAX2LIN and d-to-1 games, and show how to (approximately) solve them using efficient spectral and semidefinite program relaxations. The relaxations perform incredibly well in the presence of a large number of outlier measurements that cannot be satisfied. We use random matrix theory to prove that the algorithms almost achieve the information theoretic Shannon bound. The underlying group structure of the different applications (like SO(2), SO(3), GL(n), etc.) is heavily exploited. Applications include: cryo-electron microscopy and NMR spectroscopy for 3D protein structuring, low-rank matrix completion, clock synchronization, and surface reconstruction in computer vision and optics. Partly joint with Yoel Shkolnisky, Ronald Coifman and Fred Sigworth (Yale); Mihai Cucuringu and Yaron Lipman (Princeton); and Yosi Keller (Bar Ilan).