  We follow a stream of the history of positive matrices and positive
functionals, as applied to algebraic sums of squares decompositions, with
emphasis on the interaction between classical moment problems, function theory
of one or several complex variables and modern operator theory. The second part
of the survey focuses on recently discovered connections between real algebraic
geometry and optimization as well as polynomials in matrix variables and some
control theory problems. These new applications have prompted a series of
recent studies devoted to the structure of positivity and convexity in a free
*-algebra, the appropriate setting for analyzing inequalities on polynomials
having matrix variables. We sketch some of these developments, add to them and
comment on the rapidly growing literature.
