Polynomial optimization deals with the minimization of a polynomial function over a semialgebraic set, defined by polynomial inequalities and equations. This offers a rich model capturing optimization problems arising in various applications, ranging from combinatorial optimization, control, distance geometry, to matrix factorization ranks. We will overview the general approach introduced by Lasserre and Parrilo around 2000 for designing semidefinite approximation hierarchies, based on using sums of squares to approximate positive polynomials and the dual theory of moments. We will discuss basic properties of these hierarchies, relevant results from real algebraic geometry and moment theory, possible extensions and selected applications, in particular to the design of tractable bounds for matrix factorization ranks.