A matrix is called totally nonnegative (TNN) if each of its square submatrices has a nonnegative determinant. First seriously studied in the 1930s, TNN matrices appeared in the areas of differential equations and rational functions. In the 1950s, Karlin and MacGregor proved a probabilistic result which gave a very interesting interpretation of all TNN matrices. This interpretation led to more applications in combinatorics, algebra, electrical engineering, and chemistry. Recent work in physics and Lie theory has led to a generalization of the classical definitions, and in particular to the study of functions called totally nonnegative polynomials and Schur nonnegative polynomials. We will present a new result concerning these polynomials and will discuss how it was used in February to prove conjectures of Fomin, Leclerc, Okounkov and others. This talk will be accessible to undergraduates.