In cryo-electron microscopy (cryo-EM), a microscope generates a top view of a sample of randomly-oriented copies of a molecule. The cryo-EM problem is to use the resulting set of noisy 2D tomographic projection images taken at unknown directions to reconstruct the 3D structure of the molecule. We will discuss methods for estimating the unknown orientations using variants of semidefinite programs (SDP) that were originally proposed in the theoretical computer science community for solving problems such as Max-Cut and Unique Games. Numerical evidence suggests that the SDP method is many cases tight, that is, it provides the maximum likelihood estimator despite the fact that the parameter space is exponentially large and non-convex. If time permit, we will also discuss the problem of heterogeneity, which is the task of mapping the space of conformational states of a molecule. Here we are able to estimate the covariance matrix of the 3D structures from their 2D projections. The proposed solutions has applications beyond cryo-EM such as low-rank matrix completion and determination of ground states of interacting particle systems. The analysis combines tools from tomography, convex optimization, group theory, and random matrix theory. No prior knowledge in these areas will be assumed.