The concept of total positivity is rooted in classical mathematics where it can be traced back to works of Schoenberg on variation diminishing properties of matrices and of Gantmacher and Krein on small oscillations of mechanical systems. Since then the class of totally positive matrices and operators (and the related class of sign-regular matrices) proved to be relevant in such a wide range of applications that over the years many distinct approaches to total positivity, amenable to a particular notion, have arisen and advocated by many prominent mathematicians. This area is, however, not just a historically significant subject in mathematics, but the one that continues to produce important advances and spawn worth-wile applications. This is reflected by the topics which will be covered by the speakers of this minisymposium, viz. the study of classes of matrices related to total positivity, accurate computations based on bidiagonalization, inverse eigenvalue problems, localization of the roots of polynomials and the poles of rational functions, matrix differential equations, and Gauss quadrature rules.