  This paper presents a selected tour through the theory and applications of
lifts of convex sets. A lift of a convex set is a higher-dimensional convex set
that projects onto the original set. Many convex sets have lifts that are
dramatically simpler to describe than the original set. Finding such simple
lifts has significant algorithmic implications, particularly for optimization
problems. We consider both the classical case of polyhedral lifts, described by
linear inequalities, as well as spectrahedral lifts, defined by linear matrix
inequalities, with a focus on recent developments related to spectrahedral
lifts.
  Given a convex set, ideally we would either like to find a (low-complexity)
polyhedral or spectrahedral lift, or find an obstruction proving that no such
lift is possible. To this end, we explain the connection between the existence
of lifts of a convex set and certain structured factorizations of its
associated slack operator. Based on this characterization, we describe a
uniform approach, via sums of squares, to the construction of spectrahedral
lifts of convex sets and illustrate the method on several families of examples.
Finally, we discuss two flavors of obstruction to the existence of lifts: one
related to facial structure, and the other related to algebraic properties of
the set in question.
  Rather than being exhaustive, our aim is to illustrate the richness of the
area. We touch on a range of different topics related to the existence of
lifts, and present many examples of lifts from different areas of mathematics
and its applications.
