we prove schur positivity conjectures due to fomin-fulton-li-poon and okounkov and give several related results. we show that schur polynomials satisfy an interesting schur-log-concavity property. we discuss a "higher klyachko problem" about inequalities between littlewood-richardson coefficients and formulate a general schur positivity conjecture involving certain mysterious convex polytopes. the talk is based on a joint work with thomas lam and pavlo pylyavskyy.