  In recent years, an abundance of new molecular structures have been
elucidated using cryo-electron microscopy (cryo-EM), largely due to advances in
hardware technology and data processing techniques. Owing to these new exciting
developments, cryo-EM was selected by Nature Methods as Method of the Year
2015, and the Nobel Prize in Chemistry 2017 was awarded to three pioneers in
the field.
  The main goal of this article is to introduce the challenging and exciting
computational tasks involved in reconstructing 3-D molecular structures by
cryo-EM. Determining molecular structures requires a wide range of
computational tools in a variety of fields, including signal processing,
estimation and detection theory, high-dimensional statistics, convex and
non-convex optimization, spectral algorithms, dimensionality reduction, and
machine learning. The tools from these fields must be adapted to work under
exceptionally challenging conditions, including extreme noise levels, the
presence of missing data, and massively large datasets as large as several
Terabytes.
  In addition, we present two statistical models: multi-reference alignment and
multi-target detection, that abstract away much of the intricacies of cryo-EM,
while retaining some of its essential features. Based on these abstractions, we
discuss some recent intriguing results in the mathematical theory of cryo-EM,
and delineate relations with group theory, invariant theory, and information
theory.
