  Optimizing over the cone of nonnegative polynomials, and its dual
counterpart, optimizing over the space of moments that admit a representing
measure, are fundamental problems that appear in many different applications
from engineering and computational mathematics to business. In this paper, we
review a number of these applications. These include, but are not limited to,
problems in control (e.g., formal safety verification), finance (e.g., option
pricing), statistics and machine learning (e.g., shape-constrained regression
and optimal design), and game theory (e.g., Nash equilibria computation in
polynomial games). We then show how sum of squares techniques can be used to
tackle these problems, which are hard to solve in general. We conclude by
highlighting some directions that could be pursued to further disseminate sum
of squares techniques within more applied fields. Among other things, we
briefly address the current challenge that scalability represents for
optimization problems that involve sum of squares polynomials and discuss
recent trends in software development.
